න්‍යාස (ගණිතය) සඳහා මුලාශ්‍රය නරඹන්න

[[ගොනුව:Matrix.svg|thumb|247px|right|න්‍යාසයක කිසියම් විශේෂිත නිවේශිතයන් වෙත යොමුව දක්වන්නේ යටකුරු  අනුසාරයෙනි.]]
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න්‍යාසයක් යනු ඍජුකෝණාස්‍රාකාර ව පිළියෙළ කරන ලද සංඛ්‍යා වගුවකි.
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[[ගණිතය|ගණිතයෙහි]] '''න්‍යාසයක්''' යනු සංඛ්‍යා සමූහයක, සංකේතයන්හි හෝ ප්‍රකාශනයන්හි ඍජුකෝනාස්‍රාකාරව පිළියෙළකරන ලද වගුවකි. න්‍යාසයක අඩංගු තනි තනි අයිතම එහි මූලාවයව ලෙස හෝ ඇතුලත් කිරීම් ලෙස හඳුන්වනු ලබයි. මූලාවයව හයකින් සමන්විත න්‍යාසයක් සඳහා නිදසුනක් පහත දක්වා ඇත.
::<math>
\begin{bmatrix}
1 & 9 & -13 \\
20 & 55 & -6
\end{bmatrix}.
</math>

සමාන ප්‍රමාණයෙන් යුතු න්‍යාසයන්හි අවයව වෙන් වෙන්ව [[න්‍යාස එකතු කිරීම|එකතු කිරීමට]] හෝ අඩු කිරිමට හැකියාව ඇත. [[න්‍යාස ගුණකිරීම|න්‍යාස ගුණකිරීමේ]] නීතිය වඩාත් සංකීර්ණ වේ. පළමු න්‍යාසයෙහි සිරස් තීරු ගණනට දෙවෙනි න්‍යාසයෙහි තිරස් තීරු ගණන සමානම් පමණක් එම න්‍යාස දෙක ගුණ කිරීමට හැකියව ඇත. [[රේඛීය පරිණාමණ]] නිරූපනය කිරීම න්‍යාසයන්හි ප්‍රධාන භාවිතය වන අතර, {{nowrap|''f''(''x'') {{=}} 4''x''}} ආදී [[රේඛීය ශ්‍රිත]] සාධාරණීකරණය කිරීමටද යොදාගනියි. නිදසුනක් ලෙස, ත්‍රිමාන අවකාශය තුළ දෛශික [[දෛශික පරිවර්තනය|පරිවර්තනය]] රේඛීය පරිණාමණය වේ. R යනු [[පරිවර්තන න්‍යාසය|පරිවර්තන න්‍යාසයක්ද]] v යනු අවකාශයේ ලක්ෂ්‍යයක් විස්තර කරන [[සිරස් තීරු දෛශික|සිරස් තීරු දෛශිකයක්ද]] (එක් සිරස් තීරුවක් පමණක් ඇති න්‍යාසයක්) නම්, Rv ගුණිතය යනු එම ලක්ෂ්‍යය පරිවර්තනයෙන් පසු [[දෛශික පිහිටීම|පිහිටීම]] විස්තර කරන තීරු දෛශිකයයි. න්‍යාස දෙකක ගුණිතය රේඛීය පරිණාමණ දෙකක [[සංයෝජන ශ්‍රිත|සංයෝජනයක්]] නිරූපනයකරනු ලබයි. [[රේඛීය සමීකරණ පද්ධති|රේඛීය සමීකරණ පද්ධතියක]] විසඳුම් ලබා ගැනීම න්‍යාසයන්හි වෙනත් භාවිතයකි. න්‍යාසය [[සමචතුරස්‍රාකාර න්‍යාස|සමචතුරස්‍රාකාර]] නම් එහි [[නිර්ණායකය]] ගණනය කිරීම මඟින් එහි සමහර ගුණ අපෝහනය කිරීමට හැකියාව ඇත. නිදසුනක් ලෙස, සමචතුරස්‍රාකාර න්‍යාසයක නිර්ණායකය ශූන්‍ය නොවන්නේ නම් පමණක් එයට [[න්‍යාසයක ප්‍රතිලෝමය|ප්‍රතිලෝමය]]ක් ඇත. [[අයිගන් අගය සහ අයිගන් දෛශිකය]] රේඛීය පරිණාමණයෙහි ජ්‍යාමිතියට අන්තර් දෘශ්ටිය සලසනු ලබයි.

න්‍යාස බොහෝ විද්‍යාත්මක ක්ෂේත්‍රයන්හි භාවිත සොයාගනු ලබයි. [[භෞතික විද්‍යාව|භෞතික විද්‍යාවෙහි]], විද්‍යුත් පරිපථ, දෘශ්ටි විද්‍යාව සහ [[ක්වොන්ටම් යන්ත්‍ර  විද්‍යාව]] අධ්‍යයනය සඳහා න්‍යාස භාවිතාකරයි. [[පරිගණක ප්‍රස්තාරික ක්‍රමය|පරිගණක ප්‍රස්තාරික ක්‍රමයෙහි]], ද්විමාන තලයෙහි ත්‍රිමාන ප්‍රතිබිම්බ ප්‍රක්ෂේපනය කිරීමට සහ තාත්වික චලිතය පෙනෙන අයුරු නිර්මාණය කිරීමට න්‍යාස භාවිතාකරයි. බහු මාන සඳහා [[ව්‍යුත්පන්න]] සහ [[දර්ශක ඝාතීය]] යොදාගන්නවාසේම පෞරාණික [[ගණිතම විශ්ලේෂණය|විශ්ලේෂණාත්මක]] මත පිළිබඳ අවබෝධයක් ලබාගැනීමට [[න්‍යාස කලනය]] යොදාගනියි.

අවුරුදු සිය ගණනක් තිස්සේ වස්තු විෂය වෙමින් වර්තමානය වන විට පර්යේෂණ මට්ටම දක්වාම විහිදී ගිය න්‍යාස ගණනය සඳහා කාර්යක්ෂම ඇල්ගොරිතම සංවර්ධනය කිරීම [[සංඛ්‍යාත්මක විශ්ලේෂණය|සංඛ්‍යාත්මක විශ්ලේෂණයේද]] ප්‍රධාන කොටසක් වී පවතී. සෛද්ධාන්තිකවත් ප්‍රායෝගිකවත්, [[න්‍යාස වියෝජන ක්‍රමය]] න්‍යාස ගණනය සරල කර දෙයි. එක් එක් න්‍යාස ආකෘති උදා:- [[විරල න්‍යාස]], [[කර්ණය ආසන්න න්‍යාස]], සඳහා ඈඳන ලද ඇල්ගොරිතම [[පරිමිත මුලාවයව ක්‍රමය|පරිමිත මුලාවයව ක්‍රමයේ]] හා වෙනත් න්‍යාස ගණනයන් ඉක්මන් කරවයි. ග්‍රහ තාරකා සිද්ධාන්ත වලදී සහ පරමාණුක සිද්ධාන්ත වලදී අපරිමිත න්‍යාස භාවිතා වෙයි. ශ්‍රිතයක [[ටේලර් ශ්‍රේණිය]] මත බලපාන [[ව්‍යුත්පන්න]] කාරකය නිරූපණය කරන න්‍යාසය ඒ සඳහා ඇති හොඳම නිදසුනයි.

==අර්ථදැක්වීම==
''න්‍යාස'' යනු ගණිතමය ප්‍රකාශනයන්හි [[සෘජුකෝණාස්‍රය|සෘජුකෝණාස්‍රාකාර]] පිළියෙළ කිරීමක් වන අතර එමඟින් [[සංඛ්‍යාව|සංඛ්‍යාවල]] සරල බව ඇතිකරවිය හැකිය.<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Chapter I.1}}. Alternative references for this book include {{Harvard citations|author=Lang|year=1987b|nb=yes}} and {{Harvard citations|last1=Greub|year=1975|nb=yes}}</ref> නිදසුනක් ලෙස,

: <math>\mathbf{A} = \begin{bmatrix}

9 & 13 & 6 \\
1 & 11 & 7 \\
3 & 9 & 2 \\
6 & 0 & 7 \end{bmatrix}.</math>
විකල්ප අංකනයක් ලෙස [[කොටු වරහන|කොටු වරහන්]] වෙනුවට විශාල [[සුළු වරහන|සුළු වරහන්]] යොදාගත හැක :

: <math>\mathbf{A} = \begin{pmatrix}

9 & 13 & 6 \\
1 & 11 & 7 \\
3 & 9 & 2 \\
6 & 0 & 7 \end{pmatrix}.</math>

න්‍යාසයක තිරස් පේළි සහ සිරස් තීරු අඩංගුවේ. න්‍යාසයක අඩංගු සංඛ්‍යා එහි මූලාවයව ලෙස හෝ ඇතුලත් කිරීම් ලෙස හඳුන්වනු ලබයි. තිරස් පේළි ''m'' ගණනකින් හා සිරස් තීරු ''n'' ගණනකින් යුතු න්‍යාසයක් ''m''-by-''n'' න්‍යාසයක් ලෙස හෝ ''m''&nbsp;×&nbsp;''n'' න්‍යාසයක් ලෙස එහි ප්‍රමාණය විශේෂයෙන් සඳහන් කරන අතර, ''m'' සහ ''n'' එහි මානවේ. ඉහත සඳහන් කර ඇත්තේ 4-by-3 න්‍යාසයකි.
තිරස් පේළි එකක් (1&nbsp;×&nbsp;''n'') පමණක් ඇති න්‍යාස [[තිරස් පේළි දෛශික]] ලෙසද සිරස් තීරු එකක් (''m''&nbsp;×&nbsp;1) පමණක් ඇති න්‍යාස [[සිරස් පේළි දෛශික]] ලෙසද හඳුන්වනු ලබයි. න්‍යාසයක ඕනෑම තිරස් පේළියක් හෝ සිරස් තීරුවක් තිරස් පේළි දෛශිකය හෝ සිරස් තීරු දෛශිකය නිර්ණය කරන අතර, න්‍යාසයෙහි වෙනත් තිරස් පේළි හෝ සිරස් තීරු අනුපිළිවෙළින් ඉවත්කිරීම මඟින් ලබාගත හැකියි. නිසදසුනක් ලෙස, ඉහත '''A''' න්‍යාසයෙහි තුන්වන තිරස් පේළියේ තිරස් පේළි දෛශිකය
:<math>\begin{bmatrix}
3 & 9 & 2 \\ \end{bmatrix}.</math>
න්‍යාසයක තිරස් පේළිය හෝ සිරස් තීරුව අගයකට අර්ථපහදා දෙන විට, එය අනුරූප තිරස් පේළි දෛශිකයට හෝ සිරස් තීරු දෛශිකයට යොමුකරනු ලැබේ. උදාහරණයක් ලෙස න්‍යාසයක වෙනස් තිරස් පේළි දෙකක් සමාන බව කිව හැකියි, එහි අර්ථය ඒවායෙහි තිරස් පේළි දෛශිකය සමාන බවයි. සමහර අවස්ථාවලදී තිරස් පේළියක හෝ සිරස් තීරුවක අගය හරියටම අගයන් අනුපිළිවෙළින් ('''R'''<sup>''n''</sup> හි මූලාවයවයකඇතුලත් කිරීම් තත්වික සංඛ්‍යා නම්) න්‍යාසයට වඩා වැඩියෙන් අර්ථපහදා දිය යුතුයි, උදාරණයක් ලෙස න්‍යාසයක තිරස් පේළි අනුරූප සිරස් තීරු වලට සමාන වන විට එය එහි [[තැන් පෙරළුම් න්‍යාසය|තැන් පෙරළුම්]] න්‍යාසයවෙයි.
බොහෝ සෙයින් මෙම ලිපිය තාත්වික සහ සංකීර්ණ න්‍යාස කේන්ද්‍ර කරගෙන ඇත, තව දුරටත් න්‍යාසයක මූලාවයව කෙසේදයත් අනුපිළිවෙළින් තාත්වික හෝ සංකීර්ණ විය හැකියි. බොහෝ සාමාන්‍ය ආකාරයේ ඇතුලත් කිරීම් [[බොහෝ සාමාන්‍ය ඇතුලත් කිරීම්|පහත]] සාකච්ඡා කරනු ලබයි.

===අංකනය===
යම් දෙසකට පැතිර පවත්නා බොහෝ පුළුල් වූ න්‍යාස විශේෂීකරණ අංකනයක් ඇත. සාමාන්‍යයෙන් [[ඉංග්‍රීසි ලොකු අකුරු]] භාවිතයෙන් න්‍යාසයක් දක්වනු ලබන අතර අනුරූප [[ඉංග්‍රීසි කුඩා අකුරු]] සමඟ යටිකුරු දර්ශක දෙකක් යෙදීමෙන් ඇතුලත් කිරීමක් ඉදිරිපත් කරනු ලබයි. ඊට අමතරව ඉංග්‍රීසි ලොකු අකුරු භාවිතයෙන් න්‍යාසයක් සංකේතවත් කරනු ලබයි, බොහෝ ලේඛකයන් විශේෂ [[මුද්‍රණ ශිල්පය|මුද්‍රණ ශිල්පීය විලාසිතා]]භාවිතා කරයි, වෙනත් ගණිතමය දේවල් මඟින් න්‍යාස වල ඇති වෙනස හඳුනා ගැනීමට සුලබව සෘජු තද අකුරු (ඇල නැති) ආධාරකරනු ලබයි.  විකල්ප අංකනයක් ලෙස සෘජු අකුරු සහිතව හෝ රහිතව විචල්‍ය නාමයට යටින් ඉරි දෙකක් ගැසීම සිදුකරනු ලබයි, (e.g., <math>\underline{\underline{A}}</math>).

න්‍යාසයක ''i'' වෙනි තිරස් පේළියේ සහ ''j'' වෙනි සිරස් තීරුවේ ඇතුලත් කිරීම ''i'',''j'' වන ඇතුළත් කිරීම ලෙස සලකයි. නිදසුනක් ලෙස, ඉහත '''A''' න්‍යාසයේ (2,3) ඇතුළත් කිරිම 7 වේ. '''A''' නම් න්‍යාසයක (''i'', ''j'') ඇතුළත් කිරීම බහුල වශයෙන් භාවිත කරනුයේ ''a''<sub>''i'',''j''</sub> ලෙසිනි. '''A'''[''i,j''] හෝ '''A'''<sub>''i,j''</sub> යනු මේ සඳහා භාවිතා කරනු ලබන වෙනත් සංකේත වේ.

Sometimes a matrix is referred to by giving a formula for its (''i'',''j'')<sup>th</sup> entry, often with double parenthesis around the formula for the entry, for example, if the (''i'',''j'')<sup>th</sup> entry of '''A''' were given by ''a''<sub>''ij''</sub>, '''A''' would be denoted ((''a''<sub>''ij''</sub>)).

An asterisk is commonly used to refer to whole rows or columns in a matrix. For example, ''a''<sub>''i'',∗</sub> refers to the i<sup>th</sup> row of '''A''', and ''a''<sub>∗,''j''</sub> refers to the j<sup>th</sup> column of '''A'''. The set of all ''m''-by-''n'' matrices is denoted <math>\mathbb{M}</math>(''m'', ''n'').

A common shorthand is
:'''A''' = [''a''<sub>''i'',''j''</sub>]<sub>''i'' = 1,...,''m''; ''j'' = 1,...,''n''</sub> or more briefly '''A''' = [''a''<sub>''i'',''j''</sub>]<sub>''m''×''n''</sub>
to define an ''m'' × ''n'' matrix '''A'''. Usually the entries ''a<sub>i,j</sub>'' are defined separately for all integers {{nowrap|1 ≤ ''i'' ≤ ''m''}} and {{nowrap|1 ≤ ''j'' ≤ ''n''}}. They can however sometimes be given by one formula; for example the 3-by-4 matrix
:<math>\mathbf A = \begin{bmatrix}
0 & -1 & -2 & -3\\
1 & 0 & -1 & -2\\
2 & 1 & 0 & -1
\end{bmatrix}</math>
can alternatively be specified by '''A''' = [''i'' − ''j'']<sub>''i'' = 1,2,3; ''j'' = 1,...,4</sub>, or simply '''A''' = ((''i''-''j'')), where the size of the matrix is understood.

Some programming languages start the numbering of rows and columns at zero, in which case the entries of an ''m''-by-''n'' matrix are indexed by {{nowrap|0 ≤ ''i'' ≤ ''m'' − 1}} and {{nowrap|0 ≤ ''j'' ≤ ''n'' − 1}}.<ref>{{Harvard citations|last1=Oualline | year=2003|loc=Ch. 5|nb=yes}}</ref> This article follows the more common convention in mathematical writing where enumeration starts from 1.

==මූලික ක්‍රියාකාරකම්==
{{Main|Matrix addition|Scalar multiplication|Transpose|Row operations}}
There are a number of operations that can be applied to modify matrices called ''matrix addition'', ''scalar multiplication'' and ''transposition''.<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Definition I.2.1 (addition), Definition I.2.4 (scalar multiplication), and Definition I.2.33 (transpose)}}</ref> These form the basic techniques to deal with matrices.
{| class="wikitable"
|-
! Operation
! Definition
! Example
|-
| Addition
| The ''sum'' '''A'''+'''B''' of two ''m''-by-''n'' matrices '''A''' and '''B''' is calculated entrywise:
:('''A''' + '''B''')<sub>''i'',''j''</sub> = '''A'''<sub>''i'',''j''</sub> + '''B'''<sub>''i'',''j''</sub>, where 1 ≤ ''i'' ≤ ''m'' and 1 ≤ ''j'' ≤ ''n''.
|
<math>

\begin{bmatrix}
1 & 3 & 1 \\
1 & 0 & 0
\end{bmatrix}
+
\begin{bmatrix}
0 & 0 & 5  \\
7 & 5 & 0
\end{bmatrix}
=
\begin{bmatrix}
1+0 & 3+0 & 1+5 \\
1+7 & 0+5 & 0+0
\end{bmatrix}
=
\begin{bmatrix}
1 & 3 & 6 \\
8 & 5 & 0
\end{bmatrix}
</math>
|-
| Scalar multiplication
| The ''scalar multiplication c'''''A''' of a matrix '''A''' and a number ''c'' (also called a [[scalar (mathematics)|scalar]] in the parlance of [[abstract algebra]]) is given by multiplying every entry of '''A''' by ''c'':
:{{nowrap begin}}(''c'''''A''')<sub>''i'',''j''</sub> = ''c'' &middot; '''A'''<sub>''i'',''j''</sub>.{{nowrap end}}
| <math>2 \cdot

\begin{bmatrix}
1 & 8 & -3 \\
4 & -2 & 5
\end{bmatrix}
=
\begin{bmatrix}
2 \cdot 1 & 2\cdot 8 & 2\cdot -3 \\
2\cdot 4 & 2\cdot -2 & 2\cdot 5
\end{bmatrix}
=
\begin{bmatrix}
2 & 16 & -6 \\
8 & -4 & 10
\end{bmatrix}
</math>
|-
| Transpose
| The ''transpose'' of an ''m''-by-''n'' matrix '''A''' is the ''n''-by-''m'' matrix '''A'''<sup>''T''</sup> (also denoted '''A'''<sup>tr</sup> or <sup>t</sup>'''A''') formed by turning rows into columns and vice versa:
:{{nowrap begin}}('''A'''<sup>''T''</sup>)<sub>''i'',''j''</sub> = '''A'''<sub>''j'',''i''</sub>.{{nowrap end}}
| <math>

\begin{bmatrix}
1 & 2 & 3 \\
0 & -6 & 7
\end{bmatrix}^T =

\begin{bmatrix}
1 & 0 \\
2 & -6 \\
3 & 7
\end{bmatrix}
</math>
|}

Familiar properties of numbers extend to these operations of matrices: for example, addition is [[commutative]], i.e., the matrix sum does not depend on the order of the summands: '''A'''&nbsp;+&nbsp;'''B'''&nbsp;=&nbsp;'''B'''&nbsp;+&nbsp;'''A'''.<ref>
{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Theorem I.2.6}}
</ref>
The transpose is compatible with addition and scalar multiplication, as expressed by (''c'''''A''')<sup>''T''</sup> = ''c''('''A'''<sup>''T''</sup>) and ('''A'''&nbsp;+&nbsp;'''B''')<sup>''T''</sup>&nbsp;=&nbsp;'''A'''<sup>''T''</sup>&nbsp;+&nbsp;'''B'''<sup>''T''</sup>. Finally, ('''A'''<sup>''T''</sup>)<sup>''T''</sup>&nbsp;=&nbsp;'''A'''.
<!--Another, much less often used notion of matrix addition is the [[direct sum (matrix)|direct sum]].-->

[[Row operations]] are ways to change matrices. There are three types of row operations: row switching, that is interchanging two rows of a matrix; row multiplication, multiplying all entries of a row by a non-zero constant; and finally row addition, which means adding a multiple of a row to another row. These row operations are used in a number of ways including solving linear equations and finding inverses.

==න්‍යාස ගුණකිරීම,රේඛීය සමීකරණ සහ රේඛීය පරිණාමණ==
{{Main|Matrix multiplication}}
[[Image:Matrix multiplication diagram 2.svg|right|200px|thumb|Schematic depiction of the matrix product '''AB''' of two matrices '''A''' and '''B'''.]]
''Multiplication'' of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If '''A''' is an ''m''-by-''n'' matrix and '''B''' is an ''n''-by-''p'' matrix, then their ''matrix product'' '''AB''' is the ''m''-by-''p'' matrix whose entries are given by [[dot product]] of the corresponding row of '''A''' and the corresponding column of '''B''':

:<cite id=matrix_product><math> [\mathbf{AB}]_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + \cdots + A_{i,n}B_{n,j} = \sum_{r=1}^n A_{i,r}B_{r,j},</math></cite>

where 1 ≤ ''i'' ≤ ''m'' and 1 ≤ ''j'' ≤ ''p''.<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Definition I.2.20}}</ref> For example, the underlined entry 1 in the product is calculated as {{math|(1 × 1) + (0 × 1) + (2 × 0) {{=}} 1:}}

:<math>
\begin{align}
\begin{bmatrix}
\underline{1} & \underline 0 & \underline 2 \\
-1 & 3 & 1 \\
\end{bmatrix}

\begin{bmatrix}
3 & \underline 1 \\
2 & \underline 1 \\
1 & \underline 0 \\
\end{bmatrix}
&=
\begin{bmatrix}
5 & \underline 1 \\
4 & 2 \\
\end{bmatrix}.
\end{align}
</math>

Matrix multiplication satisfies the rules ('''AB''')'''C''' = '''A'''('''BC''') ([[associativity]]), and ('''A'''+'''B''')'''C''' = '''AC'''+'''BC''' as well as '''C'''('''A'''+'''B''') = '''CA'''+'''CB''' (left and right [[distributivity]]), whenever the size of the matrices is such that the various products are defined.<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Theorem I.2.24}}</ref> The product '''AB''' may be defined without '''BA''' being defined, namely if '''A''' and '''B''' are ''m''-by-''n'' and ''n''-by-''k'' matrices, respectively, and {{nowrap|''m'' ≠ ''k''.}} Even if both products are defined, they need not be equal, i.e., generally one has
:'''AB''' ≠ '''BA''',
i.e., <cite id=non_commutative>matrix multiplication is not [[commutativity|commutative]],</cite> in marked contrast to (rational, real, or complex) numbers whose product is independent of the order of the factors. An example of two matrices not commuting with each other is:
:<math>\begin{bmatrix}
1 & 2\\
3 & 4\\
\end{bmatrix}

\begin{bmatrix}
0 & 1\\
0 & 0\\
\end{bmatrix}=
\begin{bmatrix}
0 & 1\\
0 & 3\\
\end{bmatrix},
</math>
whereas
:<math>\begin{bmatrix}
0 & 1\\
0 & 0\\
\end{bmatrix}

\begin{bmatrix}
1 & 2\\
3 & 4\\
\end{bmatrix}=
\begin{bmatrix}
3 & 4\\
0 & 0\\
\end{bmatrix}
.</math>

The [[identity matrix]] '''I'''<sub>''n''</sub> of size ''n'' is the ''n''-by-''n'' matrix in which all the elements on the [[main diagonal]] are equal to 1 and all other elements are equal to 0, e.g.
: <math>

\mathbf{I}_3 =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}.
</math>
It is called identity matrix because multiplication with it leaves a matrix unchanged: {{nowrap begin}}'''MI'''<sub>''n''</sub> = '''I'''<sub>''m''</sub>'''M''' = '''M'''{{nowrap end}} for any ''m''-by-''n'' matrix '''M'''.

Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, such as the [[Hadamard product]] and the [[Kronecker product]].<ref>{{Harvard citations | last1=Horn | last2=Johnson |year=1985|loc=Ch. 4 and 5|nb=yes}}</ref> They arise in solving matrix equations such as the [[Sylvester equation]].

===රේඛීය සමීකරණ===
{{Main|Linear equation|System of linear equations}}
A particular case of matrix multiplication is tightly linked to ''linear equations'': if '''x''' designates a column vector (i.e., ''n''×1-matrix) of ''n'' variables ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>, and '''A''' is an ''m''-by-''n'' matrix, then the matrix equation
:'''Ax''' = '''b''',
where '''b''' is some ''m''×1-column vector, is equivalent to the system of linear equations
:''A''<sub>1,1</sub>''x''<sub>1</sub> + ''A''<sub>1,2</sub>''x''<sub>2</sub> + ... + ''A''<sub>1,''n''</sub>''x''<sub>''n''</sub> = ''b''<sub>1</sub>
:...
:''A''<sub>''m'',1</sub>''x''<sub>1</sub> + ''A''<sub>''m'',2</sub>''x''<sub>2</sub> + ... + ''A''<sub>''m'',''n''</sub>''x''<sub>''n''</sub> = ''b''<sub>''m''</sub> .<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=I.2.21 and 22}}</ref>
This way, matrices can be used to compactly write and deal with multiple linear equations, i.e., systems of linear equations.

===රේඛීය පරිණාමණ===
{{Main|Linear transformation|Transformation matrix}}
[[Image:Area parallellogram as determinant.svg|thumb|right|The vectors represented by a 2-by-2 matrix correspond to the sides of a unit square transformed into a parallelogram.]]
Matrices and matrix multiplication reveal their essential features when related to ''linear transformations'', also known as ''linear maps''. <cite id=linear_maps>A real ''m''-by-''n'' matrix '''A''' gives rise to a linear transformation '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup> mapping each vector '''x''' in '''R'''<sup>''n''</sup> to the (matrix) product '''Ax''', which is a vector in '''R'''<sup>''m''</sup>. Conversely, each linear transformation ''f'': '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup> arises from a unique ''m''-by-''n'' matrix '''A''': explicitly, the {{nowrap|(''i'', ''j'')-entry}} of '''A''' is the ''i''<sup>th</sup> coordinate of ''f''('''e'''<sub>''j''</sub>), where {{nowrap begin}}'''e'''<sub>''j''</sub> = (0,...,0,1,0,...,0){{nowrap end}} is the [[unit vector]] with 1 in the ''j''<sup>th</sup> position and 0 elsewhere.</cite> The matrix '''A''' is said to represent the linear map ''f'', and '''A''' is called the ''transformation matrix'' of ''f''.

For example, the 2&times;2 matrix

:<math>
\mathbf A = \begin{bmatrix} a & c\\b & d \end{bmatrix}\,
</math>

can be viewed as the transform of the [[unit square]] into a [[parallelogram]] with vertices at {{nowrap|(0, 0)}}, {{nowrap|(''a'', ''b'')}}, {{nowrap|(''a'' + ''c'', ''b'' + ''d'')}}, and {{nowrap|(''c'', ''d'')}}. The parallelogram pictured at the right is obtained by multiplying '''A''' with each of the column vectors <math>\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} </math> and <math>\begin{bmatrix}0 \\ 1\end{bmatrix}</math> in turn. These vectors define the vertices of the unit square. 

The following table shows a number of [[2 × 2 real matrices|2-by-2 matrices]] with the associated linear maps of '''R'''<sup>2</sup>. The blue original is mapped to the green grid and shapes, the origin (0,0) is marked with a black point.
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
|-
| [[Shear mapping|Horizontal shear]] with m=1.25.
| Horizontal flip
| [[Squeeze mapping]] with r=3/2
|[[Scaling (geometry)|Scaling]] by a factor of 3/2
|<cite id=rotation_matrix>[[Rotation matrix|Rotation]] by π/6<sup>R</sup> = 30°</cite>
|-
| <math>\begin{bmatrix}
1 & 1.25  \\
0 & 1 \end{bmatrix}</math>
| <math>\begin{bmatrix}
-1 & 0  \\
0 & 1 \end{bmatrix}</math>
| <math>\begin{bmatrix}
3/2 & 0  \\
0 & 2/3 \end{bmatrix}</math>
|<math>\begin{bmatrix}
3/2 & 0  \\
0 & 3/2 \end{bmatrix}</math>
|<math>\begin{bmatrix}\cos(\pi / 6^{R}) & -\sin(\pi / 6^{R})\\ \sin(\pi / 6^{R}) & \cos(\pi / 6^{R})\end{bmatrix}</math>
|-
|width=20%| [[Image:VerticalShear m=1.25.svg|175px]]
|width=20%| [[Image:Flip map.svg|150px]]
|width=20%| [[Image:Squeeze r=1.5.svg|150px]]
|width=20%| [[Image:Scaling by 1.5.svg|125px]]
|width=20%| [[Image:Rotation by pi over 6.svg|125px]]
|}

Under the [[bijection|1-to-1 correspondence]] between matrices and linear maps, matrix multiplication corresponds to [[function composition|composition]] of maps:<ref>{{Harvard citations | last1=Greub | year=1975 |nb=yes|loc=Section III.2}}</ref> if a ''k''-by-''m'' matrix '''B''' represents another linear map ''g'' : '''R'''<sup>''m''</sup> → '''R'''<sup>''k''</sup>, then the composition {{nowrap|''g'' ∘ ''f''}} is represented by '''BA''' since
:(''g'' ∘ ''f'')('''x''') = ''g''(''f''('''x''')) = ''g''('''Ax''') = '''B'''('''Ax''') = ('''BA''')'''x'''.
The last equality follows from the above-mentioned associativity of matrix multiplication.

The [[rank of a matrix]] '''A''' is the maximum number of [[linear independence|linearly independent]] row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors.<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Definition II.3.3}}</ref> Equivalently it is the [[Hamel dimension|dimension]] of the [[Image (mathematics)|image]] of the linear map represented by '''A'''.<ref>{{Harvard citations | last1=Greub | year=1975 |nb=yes|loc=Section III.1}}</ref> The [[rank-nullity theorem]] states that the dimension of the [[kernel (matrix)|kernel]] of a matrix plus the rank equals the number of columns of the matrix.<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Theorem II.3.22}}</ref>

==සමචතුරස්‍රාකාර න්‍යාස==<!-- This section is linked from [[Orthogonal matrix]] -->
A ''square matrix'' is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order ''n.'' Any two square matrices of the same order can be added and multiplied. A square matrix '''A''' is called ''[[invertible matrix|invertible]]'' or ''non-singular'' if there exists a matrix '''B''' such that
: '''AB''' = '''I'''<sub>''n''</sub>.<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Definition I.2.28}}</ref>
This is equivalent to '''BA''' = '''I'''<sub>''n''</sub>.<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Definition I.5.13}}</ref> Moreover, if '''B''' exists, it is unique and is called the ''[[inverse matrix]]'' of '''A''', denoted '''A'''<sup>−1</sup>.

The entries A<sub>''i'',''i''</sub> form the [[main diagonal]] of a matrix. The [[trace of a matrix|trace]], tr('''A''') of a square matrix '''A''' is the sum of its diagonal entries. While, as mentioned [[#non commutative|above]], matrix multiplication is not commutative, the trace of the product of two matrices is independent of the order of the factors: tr('''AB''') = tr('''BA''').<ref>This is immediate from the definition of matrix multiplication.
:<math>\scriptstyle\operatorname{tr}(\mathsf{AB}) = \sum_{i=1}^m \sum_{j=1}^n A_{ij} B_{ji} = \operatorname{tr}(\mathsf{BA}).</math></ref>

Also, the trace of a matrix is equal to that of its transpose, i.e., {{nowrap begin}}tr('''A''') = tr('''A'''<sup>T</sup>){{nowrap end}}.

If all entries outside the main diagonal are zero, '''A''' is called a [[diagonal matrix]]. If only all entries above (below) the main diagonal are zero, '''A''' is called a lower [[triangular matrix]] (upper triangular matrix, respectively). For example, if ''n'' = 3, they look like

:<math>
      \begin{bmatrix}
           d_{11} & 0 & 0 \\
           0 & d_{22} & 0 \\
           0 & 0 & d_{33} \\
        \end{bmatrix}
</math> (diagonal), <math>
      \begin{bmatrix}
           l_{11} & 0 & 0 \\
           l_{21} & l_{22} & 0 \\
           l_{31} & l_{32} & l_{33} \\
        \end{bmatrix}
</math> (lower) and <math>
        \begin{bmatrix}
           u_{11} & u_{12} & u_{13} \\
           0 & u_{22} & u_{23} \\
           0 & 0 & u_{33} \\
        \end{bmatrix}</math>  (upper triangular matrix).

===නිර්ණායකය===
{{Main|Determinant}}
[[Image:Determinant Example.png|thumb|300px|right|A linear transformation on '''R'''<sup>2</sup> given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the [[orientation (mathematics)|orientation]], since it turns the counterclockwise orientation of the vectors to a clockwise one.]]

The ''determinant'' det('''A''') or |'''A'''| of a square matrix '''A''' is a number encoding certain properties of the matrix. A matrix is invertible [[if and only if]] its determinant is nonzero. Its [[absolute value]] equals the area (in '''R'''<sup>2</sup>) or volume (in '''R'''<sup>3</sup>) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved. 

The determinant of 2-by-2 matrices is given by
:<math>\det \begin{pmatrix}a&b\\c&d\end{pmatrix} = ad-bc.</math>
When the determinant is equal to one, then the matrix represents an [[2 × 2 real matrices#Equi-areal mapping|equi-areal mapping]]. The determinant of 3-by-3 matrices involves 6 terms ([[rule of Sarrus]]). The more lengthy [[Leibniz formula for determinants|Leibniz formula]] generalises these two formulae to all dimensions.<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Definition III.2.1}}</ref>

The determinant of a product of square matrices equals the product of their determinants: {{nowrap begin}}det('''AB''') = det('''A''') · det('''B''').<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Theorem III.2.12}}</ref>{{nowrap end}} Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1.<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Corollary III.2.16}}</ref> Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the [[Laplace expansion]] expresses the determinant in terms of [[minor (linear algebra)|minors]], i.e., determinants of smaller matrices.<ref>{{Harvard citations|last1 =  Mirsky|year = 1990|nb = yes|loc=Theorem 1.4.1}}</ref> This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve [[linear system]]s using [[Cramer's rule]], where the division of the determinants of two related square matrices equates to the value of each of the system's variables.<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Theorem III.3.18}}</ref>

===අයිගන් අගයන් සහ අයිගන් දෛශික===
{{Main|Eigenvalue, eigenvector and eigenspace|l1=Eigenvalues and eigenvectors}}
A number λ and a non-zero vector '''v''' satisfying
:'''Av''' = λ'''v'''
are called an ''eigenvalue'' and an ''eigenvector'' of '''A''', respectively.<ref group=nb>''Eigen'' means "own" in [[ජර්මානු භාෂාව|ජර්මානු]] and in [[Dutch language|Dutch]].</ref><ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Definition III.4.1}}</ref> The number λ is an eigenvalue of an ''n''×''n''-matrix '''A''' if and only if '''A'''−λ'''I'''<sub>''n''</sub> is not invertible, which is [[logical equivalence|equivalent]] to
:<math>\det(\mathsf{A}-\lambda \mathsf{I}) = 0.\ </math><ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Definition III.4.9}}</ref>
The function ''p''<sub>'''A'''</sub>(''t'') = det('''A'''−''t'''''I''') is called the [[characteristic polynomial]] of '''A''', its [[degree of a polynomial|degree]] is ''n''. Therefore ''p''<sub>'''A'''</sub>(''t'') has at most ''n'' different roots, i.e., eigenvalues of the matrix.<ref>{{Harvard citations|last1 =  Brown|year = 1991|nb = yes|loc=Corollary III.4.10}}</ref> They may be complex even if the entries of '''A''' are real. According to the [[Cayley–Hamilton theorem]], {{nowrap begin}}''p''<sub>'''A'''</sub>('''A''') = '''0'''{{nowrap end}}, that is to say, the characteristic polynomial applied to the matrix itself yields the [[zero matrix]].

===සමමිතිය===
A square matrix '''A''' that is equal to its transpose, i.e., {{nowrap begin}}'''A''' = '''A'''<sup>T</sup>{{nowrap end}}, is a [[symmetric matrix]].  If instead, '''A''' was equal to the negative of its transpose, i.e., {{nowrap begin}}'''A''' = −'''A'''<sup>T</sup>,{{nowrap end}} then '''A''' is a [[skew-symmetric matrix]]. In complex matrices, symmetry is often replaced by the concept of [[Hermitian matrix|Hermitian matrices]], which satisfy '''A'''<sup>∗</sup> = '''A''', where the star or [[asterisk]] denotes the [[conjugate transpose]] of the matrix, i.e., the transpose of the [[complex conjugate]] of '''A'''.

By the [[spectral theorem]], real symmetric matrices and complex Hermitian matrices have an [[eigenbasis]]; i.e., every vector is expressible as a [[linear combination]] of eigenvectors. In both cases, all eigenvalues are real.<ref>{{Harvard citations|last1 =  Horn|last2=Johnson|year = 1985|nb = yes|loc=Theorem 2.5.6}}</ref> This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see [[#Infinite matrices|below]].

===නිශ්චිත භාවය===
{| class="wikitable" style="float:right; text-align:center; margin:0ex 0ex 1ex 1ex;"
|-
|  colspan=2 | Matrix '''A'''; definiteness; associated quadratic form ''Q''<sub>'''A'''</sub>(''x'',''y''); <br /> set of vectors (''x'',''y'') such that ''Q''<sub>'''A'''</sub>(''x'',''y'')=1
|-
| <math>\begin{bmatrix}
1/4 & 0\\
0 & 1/4\end{bmatrix}</math> || <math>\begin{bmatrix}
1/4 & 0\\
0 & -1/4\end{bmatrix}</math>
|-
| positive definite || indefinite
|-
| 1/4 ''x''<sup>2</sup> + 1/4''y''<sup>2</sup> || 1/4 ''x''<sup>2</sup> − 1/4 ''y''<sup>2</sup>
|-
| [[Image:Ellipse in coordinate system with semi-axes labelled.svg|150px]] <br />[[Ellipse]] || [[Image:Hyperbola2.png|100px]]<br />[[Hyperbola]]
|}
A symmetric ''n''×''n''-matrix is called'' [[positive-definite matrix|positive-definite]]'' (respectively negative-definite; indefinite), if for all nonzero vectors '''x'''&nbsp;∈&nbsp;'''R'''<sup>''n''</sup> the associated [[quadratic form]] given by
:<cite id=quadratic_forms>''Q''('''x''') = '''x'''<sup>T</sup>'''Ax'''</cite>
takes only positive values (respectively only negative values; both some negative and some positive values).<ref>{{Harvard citations|last1 =  Horn|last2=Johnson|year = 1985|nb = yes|loc=Chapter 7}}</ref> If the quadratic form takes only non-negative (respectively only non-positive) values, the symmetric matrix is called positive-semidefinite (respectively negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.

A symmetric matrix is positive-definite if and only if all its eigenvalues are positive.<ref>{{Harvard citations|last1 =  Horn|last2=Johnson|year = 1985|nb = yes|loc=Theorem 7.2.1}}</ref> The table at the right shows two possibilities for 2-by-2 matrices.

Allowing as input two different vectors instead yields the [[bilinear form]] associated to '''A''':
:''B''<sub>'''A'''</sub> ('''x''', '''y''') = '''x'''<sup>T</sup>'''Ay'''.<ref>{{Harvard citations|last1 =  Horn|last2=Johnson|year = 1985|nb = yes|loc=Example 4.0.6, p. 169}}</ref>

==සංඛ්‍යාත්මක ආකාර==
In addition to theoretical knowledge of properties of matrices and their relation to other fields, it is important for practical purposes to perform matrix calculations effectively and precisely. The domain studying these matters is called [[numerical linear algebra]].<ref>{{Harvard citations|last1 =  Bau III|last2=Trefethen|year = 1997|nb = yes}}</ref> As with other numerical situations, two main aspects are the [[complexity analysis|complexity of algorithms]] and their [[numerical stability]]. Many problems can be solved by both direct algorithms or iterative approaches. For example, finding eigenvectors can be done by finding a [[sequence (mathematics)|sequence]] of vectors '''x'''<sub>''n''</sub> [[Limit of a sequence|converging]] to an eigenvector when ''n'' tends to [[infinity]].<ref>{{Harvard citations | last1=Householder|year=1975|loc=Ch. 7|nb=yes}}</ref>

Determining the complexity of an algorithm means finding [[upper bound]]s or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, e.g., multiplication of matrices. For example, calculating the matrix product of two ''n''-by-''n'' matrix using the definition given above needs ''n''<sup>3</sup> multiplications, since for any of the ''n''<sup>2</sup> entries of the product, ''n'' multiplications are necessary. The [[Strassen algorithm]] outperforms this "naive" algorithm; it needs only ''n''<sup>2.807</sup> multiplications.<ref>{{Harvard citations|last1 =  Golub|last2=Van Loan|year = 1996|nb = yes|loc=Algorithm 1.3.1}}</ref> A refined approach also incorporates specific features of the computing devices.

In many practical situations additional information about the matrices involved is known. An important case are [[sparse matrix|sparse matrices]], i.e., matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems '''Ax''' = '''b''' for sparse matrices '''A''', such as the [[conjugate gradient method]].<ref>{{Harvard citations|last1 =  Golub|last2=Van Loan|year = 1996|nb = yes|loc=Chapters 9 and 10, esp. section 10.2}}</ref>

An algorithm is, roughly speaking, numerically stable, if little deviations (such as rounding errors) do not lead to big deviations in the result. For example, calculating the inverse of a matrix via [[Laplace's formula]] (Adj ('''A''') denotes the [[adjugate matrix]] of '''A''')
:'''A'''<sup>−1</sup> = Adj('''A''') / det('''A''')
may lead to significant rounding errors if the determinant of the matrix is very small. The [[matrix norm|norm of a matrix]] can be used to capture the [[Condition number|conditioning]] of linear algebraic problems, such as computing a matrix' inverse.<ref>{{Harvard citations|last1 =  Golub|last2=Van Loan|year = 1996|nb = yes|loc=Chapter 2.3}}</ref>

Although most [[computer language]]s are not designed with commands or libraries for matrices, as early as the 1970s, some engineering desktop computers such as the [[HP 9830]] had ROM cartridges to add BASIC commands for matrices. Some computer languages such as [[APL (programming language)|APL]] were designed to manipulate matrices, and [[List of numerical analysis software|various mathematical programs]] can be used to aid computing with matrices.<ref>For example, [[Mathematica]], see {{Harvard citations | last1=Wolfram  |year=2003|loc=Ch. 3.7|nb=yes}}</ref>

===න්‍යාස වියෝජන ක්‍රම===
{{Main|Matrix decomposition|Matrix diagonalization|Gaussian elimination|Montante's method}}
There are several methods to render matrices into a more easily accessible form. They are generally referred to as ''matrix transformation'' or ''matrix decomposition'' techniques. The interest of all these decomposition techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices.

The [[LU decomposition]] factors matrices as a product of lower ('''L''') and an upper [[triangular matrix|triangular matrices]] ('''U''').<ref>{{Harvard citations | last1=Press | last2=Flannery | last3=Teukolsky |year=1992|nb=yes}}</ref> Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called [[forward substitution|forward and back substitution]]. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The ''Gaussian elimination'' is a similar algorithm; it transforms any matrix to [[row echelon form]].<ref>{{Harvard citations | last1=Stoer | last2=Bulirsch |year=2002|nb=yes|loc=Section 4.1}}</ref> Both methods proceed by multiplying the matrix by suitable [[elementary matrix|elementary matrices]], which correspond to [[permutation matrix|permuting rows or columns]] and adding multiples of one row to another row. [[Singular value decomposition]] expresses any matrix '''A''' as a product '''UDV'''<sup>∗</sup>, where '''U''' and '''V''' are [[unitary matrix|unitary matrices]] and '''D''' is a diagonal matrix.

[[File:Jordan blocks.svg|right|thumb|A matrix in Jordan normal form. The grey blocks are called Jordan blocks.]]
The [[eigendecomposition]] or ''diagonalization'' expresses '''A''' as a product '''VDV'''<sup>−1</sup>, where '''D''' is a diagonal matrix and '''V''' is a suitable invertible matrix.<ref>{{Harvard citations|last1 =  Horn|last2=Johnson|year = 1985|nb = yes|loc=Theorem 2.5.4}}</ref>  If '''A''' can be written in this form, it is called [[diagonalizable matrix|diagonalizable]]. More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into [[Jordan normal form]], that is to say matrices whose only nonzero entries are the eigenvalues λ<sub>1</sub> to λ<sub>n</sub> of '''A''', placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right.<ref>{{Harvard citations|last1 =  Horn|last2=Johnson|year = 1985|nb = yes|loc=Ch. 3.1, 3.2}}</ref>  Given the eigendecomposition, the ''n''<sup>th</sup> power of '''A''' (i.e., ''n''-fold iterated matrix multiplication) can be calculated via
:'''A'''<sup>''n''</sup> = ('''VDV'''<sup>−1</sup>)<sup>''n''</sup> = '''VDV'''<sup>−1</sup>'''VDV'''<sup>−1</sup>...'''VDV'''<sup>−1</sup> = '''VD'''<sup>''n''</sup>'''V'''<sup>−1</sup>
and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for '''A''' instead. This can be used to compute the [[matrix exponential]] ''e''<sup>'''A'''</sup>, a need frequently arising in solving [[linear differential equation]]s, [[matrix logarithm]]s and [[square root of a matrix|square roots of matrices]].<ref>{{Harvard citations|last1 =  Arnold|last2=Cooke|year=1992|nb=yes|loc=Sections 14.5, 7, 8}}</ref> To avoid numerically [[condition number|ill-conditioned]] situations, further algorithms such as the [[Schur decomposition]] can be employed.<ref>{{Harvard citations | last1=Bronson |year=1989|nb=yes|loc=Ch. 15}}</ref>

==වීජීය සංක්ෂේපන ආකාරය සහ සාධාරණීකරණය==
Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general [[Field (mathematics)|fields]] or even [[Ring (mathematics)|rings]], while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension are [[tensors]], which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or two-dimensional array of numbers.<ref>{{Harvard citations|last1 =  Coburn|year = 1955|nb = yes|loc=Ch. V}}</ref> Matrices, subject to certain requirements tend to form [[group (mathematics)|groups]] known as matrix groups.

===න්‍යාස සමඟ බොහෝ සාමාන්‍ය ඇතුලත් කිරීම්===
This article focuses on matrices whose entries are real or [[complex number]]s. <cite id=more_general_entries>However, matrices can be considered with much more general types of entries than real or complex numbers.</cite> As a first step of generalization, any [[field (mathematics)|field]], i.e., a [[set (mathematics)|set]] where [[addition]], [[subtraction]], [[multiplication]] and [[Division (mathematics)|division]] operations are defined and well-behaved, may be used instead of '''R''' or '''C''', for example [[rational number]]s or [[finite field]]s. For example, [[coding theory]] makes use of matrices over finite fields. Wherever [[eigenvalue]]s are considered, as these are roots of a polynomial they may exist only in a larger field than that of the coefficients of the matrix; for instance they may be complex in case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field (e.g., to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues. Alternatively one can consider only matrices with entries in an [[algebraically closed field]], such as '''C''', from the outset.

More generally, abstract algebra makes great use of matrices with entries in a [[ring (mathematics)|ring]] ''R''.<ref>{{Harvard citations|last1 =  Lang|year = 2002|nb = yes|loc=Chapter XIII}}</ref> Rings are a more general notion than fields in that no division operation exists. The very same addition and multiplication operations of matrices extend to this setting, too. The set M(''n'', ''R'') of all square ''n''-by-''n'' matrices over ''R'' is a ring called [[matrix ring]], isomorphic to the [[endomorphism ring]] of the left ''R''-[[module (mathematics)|module]] ''R''<sup>''n''</sup>.<ref>{{Harvard citations|last1 =  Lang|year = 2002|nb =yes|loc=XVII.1, p. 643}}</ref> If the ring ''R'' is [[commutative ring|commutative]], i.e., its multiplication is commutative, then M(''n'', ''R'') is a unitary noncommutative (unless ''n'' = 1) [[associative algebra]] over ''R''. The [[determinant]] of square matrices over a commutative ring ''R'' can still be defined using the [[Leibniz formula (determinant)|Leibniz formula]]; such a matrix is invertible if and only if its determinant is [[invertible]] in ''R'', generalising the situation over a field ''F'', where every nonzero element is invertible.<ref>{{Harvard citations|last1 =  Lang|year = 2002|nb =yes|loc=Proposition XIII.4.16}}</ref> Matrices over [[superring]]s are called [[supermatrix|supermatrices]].<ref>{{Harvard citations|last1 =  Reichl|year=2004|nb=yes|loc=Section L.2}}</ref>

Matrices do not always have all their entries in the same ring – or even in any ring at all. One special but common case is [[block matrix|block matrices]], which may be considered as matrices whose entries themselves are matrices. The entries need not be quadratic matrices, and thus need not be members of any ordinary [[Ring (mathematics)|ring]]; but their sizes must fulfil certain compatibility conditions.

===රේඛීය සිතියම් වලට ඇති සබැඳියාව===
Linear maps '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup> are equivalent to ''m''-by-''n'' matrices, as described [[#linear maps|above]]. More generally, any linear map {{nowrap|''f'': ''V'' → ''W''}} between finite-[[Hamel dimension|dimensional]] [[vector space]]s can be described by a matrix '''A''' = (''a''<sub>''ij''</sub>), after choosing [[basis (linear algebra)|bases]] '''v'''<sub>1</sub>, ..., '''v'''<sub>''n''</sub> of ''V'', and '''w'''<sub>1</sub>, ..., '''w'''<sub>''m''</sub> of ''W'' (so ''n'' is the dimension of ''V'' and ''m'' is the dimension of ''W''), which is such that
:<math>f(\mathbf{v}_j) = \sum_{i=1}^m a_{i,j} \mathbf{w}_i\qquad\mbox{for }j=1,\ldots,n.</math>
In other words, column ''j'' of ''A'' expresses the image of '''v'''<sub>''j''</sub> in terms of the basis vectors '''w'''<sub>''i''</sub> of ''W''; thus this relation uniquely determines the entries of the matrix '''A'''. Note that the matrix depends on the choice of the bases: different choices of bases give rise to different, but [[Matrix equivalence|equivalent matrices]].<ref>{{Harvard citations | last1=Greub | year=1975 |nb=yes|loc=Section III.3}}</ref> Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix '''A'''<sup>''T''</sup> describes the [[Transpose of a linear map|transpose of the linear map]] given by '''A''', with respect to the [[dual space|dual bases]].<ref>{{Harvard citations | last1=Greub | year=1975 |nb=yes|loc=Section III.3.13}}</ref>

More generally, the set of ''m''×''n'' matrices can be used to represent the ''R''-linear maps between the free modules ''R''<sup>''m''</sup> and ''R''<sup>''n''</sup> for an arbitrary ring ''R'' with unity.  When ''n''&nbsp;=&nbsp;''m'' composition of these maps is possible, and this gives rise to the [[matrix ring]] of ''n''×''n'' matrices representing the [[endomorphism ring]] of ''R''<sup>''n''</sup>.

===න්‍යාස කාණ්ඩ===
{{Main|Matrix group}}
A [[group (mathematics)|group]] is a mathematical structure consisting of a set of objects together with a [[binary operation]], i.e., an operation combining any two objects to a third, subject to certain requirements.<ref>See any standard reference in group.</ref> A group in which the objects are matrices and the group operation is matrix multiplication is called a ''matrix group''.<ref group=nb>Additionally, the group is required to be [[closed subset|closed]] in the general linear group.</ref><ref>{{Harvard citations|last1 =  Baker|year = 2003|nb = yes|loc=Def. 1.30}}</ref> Since in a group every element has to be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the [[general linear group]]s.

Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups. For example, matrices with a given size and with a determinant of 1 form a [[subgroup]] of (i.e., a smaller group contained in) their general linear group, called a [[special linear group]].<ref>{{Harvard citations|last1 =  Baker|year = 2003|nb = yes|loc=Theorem 1.2}}</ref> [[orthogonal matrix|Orthogonal matrices]], determined by the condition
:'''M'''<sup>T</sup>'''M''' = '''I''',
form the [[orthogonal group]].<ref>{{Harvard citations|last1 =  Artin|year = 1991|nb = yes|loc=Chapter 4.5}}</ref>  They are called ''orthogonal'' since the associated linear transformations of '''R'''<sup>''n''</sup> preserve angles in the sense that the [[scalar product]] of two vectors is unchanged after applying '''M''' to them:
:('''Mv''') &middot; ('''Mw''') = '''v''' &middot; '''w'''.<ref>{{Harvard citations|last1 =  Artin|year = 1991|nb = yes|loc=Theorem 4.5.13}}</ref>
Every [[finite group]] is [[isomorphic]] to a matrix group, as one can see by considering the [[regular representation]] of the [[symmetric group]].<ref>{{Harvard citations | last1=Rowen| year=2008|loc=Example 19.2, p. 198|nb=yes}}</ref> General groups can be studied using matrix groups, which are comparatively well-understood, by means of [[representation theory]].<ref>See any reference in representation theory or [[group representation]].</ref>

===අපරිමිත න්‍යාස===
It is also possible to consider matrices with infinitely many rows and/or columns<ref>See the item "Matrix" in {{Harvard citations | editor1-last=Itõ | year=1987 |nb=yes}}</ref> even if, being infinite objects, one cannot write down such matrices explicitly. All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a well-defined entry (these index sets need not even be subsets of the natural numbers). The basic operations of addition, subtraction, scalar multiplication and transposition can still be defined without problem; however matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general.

If ''R'' is any ring with unity, then the ring of endomorphisms of <math>M=\bigoplus_{i\in I}R</math> as a right ''R'' module is isomorphic to the ring of '''column finite matrices''' <math>\mathbb{CFM}_I(R)</math> whose entries are indexed by <math>I\times I</math>, and whose columns each contain only finitely many nonzero entries.  The endomorphisms of ''M'' considered as a left ''R'' module result in an analogous object, the '''row finite matrices''' <math>\mathbb{RFM}_I(R)</math> whose rows each only have finitely many nonzero entries.

If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason. For a matrix '''A''' to describe a linear map ''f'': ''V''→''W'', bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) [[linear combination]] of basis vectors, so that written as a (column) vector&nbsp;''v'' of coefficients, only finitely many entries ''v''<sub>''i''</sub> are nonzero. Now the columns of '''A''' describe the images by ''f'' of individual basis vectors of ''V'' in the basis of ''W'', which is only meaningful if these columns have only finitely many nonzero entries. There is no restriction on the rows of ''A'' however: in the product '''A'''·''v'' there are only finitely many nonzero coefficients of ''v'' involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined. Moreover this amounts to forming a linear combination of the columns of '''A''' that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries, because each of those columns do. One also sees that products of two matrices of the given type is well defined (provided as usual that the column-index and row-index sets match), is again of the same type, and corresponds to the composition of linear maps.

If ''R'' is a [[normed ring]], then the condition of row or column finiteness can be relaxed.  With the norm in place, [[absolutely convergent series]] can be used instead of finite sums.  For example, the matrices whose column sums are absolutely convergent sequences form a ring.  Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring.  

In that vein, infinite matrices can also be used to describe [[Hilbert space#Operators on Hilbert spaces|operators on Hilbert spaces]], where convergence and [[Continuous function|continuity]] questions arise, which again results in certain constraints that have to be imposed. However, the explicit point of view of matrices tends to obfuscate the matter,<ref group=nb>"Not much of matrix theory carries over to infinite-dimensional spaces, and what does is not so useful, but it sometimes helps." {{Harvard citations | last1=Halmos |year=1982 | loc=p. 23, Chapter 5|nb=yes}}</ref> and the abstract and more powerful tools of [[functional analysis]] can be used instead.

===ශූන්‍ය න්‍යාස===
An ''empty matrix'' is a matrix in which the number of rows or columns (or both) is zero.<ref>"Empty Matrix: A matrix is empty if either its row or column dimension is zero", [http://www.omatrix.com/manual/glossary.htm Glossary] {{Webarchive|url=https://web.archive.org/web/20090429015728/http://omatrix.com/manual/glossary.htm |date=2009-04-29 }}, O-Matrix v6 User Guide</ref><ref>"A matrix having at least one dimension equal to zero is called an empty matrix", [http://www.system.nada.kth.se/unix/software/matlab/Release_14.1/techdoc/matlab_prog/ch_dat29.html MATLAB Data Structures] {{Webarchive|url=https://web.archive.org/web/20091228102653/http://www.system.nada.kth.se/unix/software/matlab/Release_14.1/techdoc/matlab_prog/ch_dat29.html |date=2009-12-28 }}</ref> Empty matrices help dealing with maps involving the [[zero vector space]]. For example, if ''A'' is a 3-by-0 matrix ''A'' and ''B'' is a 0-by-3 matrix, then ''AB'' is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space ''V'' to itself, while ''BA'' is a 0-by-0 matrix. There is no common notation for empty matrices, but most [[computer algebra system]]s allow creating and computing with them. The determinant of the 0-by-0 matrix is 1 as follows from regarding the [[empty product]] occurring in the Leibniz formula for the determinant as 1. This value is also consistent with the fact that the identity map from any finite dimensional space to itself has determinant&nbsp;1, a fact that is often used as a part of the characterization of determinants.

==භාවිතය==
There are numerous applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix. For example, in [[game theory]] and [[economics]], the [[payoff matrix]] encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose.<ref>{{Harvard citations|last1=Fudenberg | last2=Tirole|year=1983|nb=yes|loc=Section 1.1.1}}</ref> [[Text mining]] and automated [[thesaurus]] compilation makes use of [[document-term matrix|document-term matrices]] such as [[tf-idf]] to track frequencies of certain words in several documents.<ref>{{Harvard citations | last1=Manning |year=1999|loc=Section 15.3.4|nb=yes}}</ref>

Complex numbers can be represented by particular real 2-by-2 matrices via
:<math>a + ib \leftrightarrow \begin{bmatrix}
a & -b  \\
b & a \end{bmatrix},</math>
under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of [[absolute value]] 1, as [[#rotation matrix|above]]. A similar interpretation is possible for [[quaternion]]s.<ref>{{Harvard citations | last1=Ward  | year=1997 | loc=Ch. 2.8|nb=yes}}</ref>

Early [[encryption]] techniques such as the [[Hill cipher]] also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break.<ref>{{Harvard citations | last1=Stinson|year=2005|loc=Ch. 1.1.5 and 1.2.4|nb=yes}}</ref> [[Computer graphics]] uses matrices both to represent objects and to calculate transformations of objects using affine [[rotation matrix|rotation matrices]] to accomplish tasks such as projecting a three-dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation.<ref>{{Harvard citations | last1=Association for Computing Machinery |year=1979|loc=Ch. 7|nb=yes}}</ref> Matrices over a [[polynomial ring]] are important in the study of [[control theory]].

[[Chemistry]] makes use of matrices in various ways, particularly since the use of [[Quantum mechanics|quantum theory]] to discuss [[chemical bond|molecular bonding]] and [[spectroscopy]]. Examples are the [[overlap matrix]] and the [[Fock matrix]] used in solving the [[Roothaan equations]] to obtain the [[molecular orbital]]s of the [[Hartree–Fock]] method.

===ප්‍රස්තාරික සිද්ධාන්තය===
[[Image:Labelled undirected graph.svg|150px|thumb|right|An undirected graph with adjacency matrix <math>\begin{bmatrix}
2 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0
\end{bmatrix}.</math>]]
The [[adjacency matrix]] of a [[finite graph]] is a basic notion of [[graph theory]].<ref>{{Harvard citations|last1 =  Godsil|last2=Royle|year = 2004|nb = yes|loc=Ch. 8.1}}</ref> It saves which vertices of the graph are connected by an edge. Matrices containing just two different values (0 and 1 meaning for example "yes" and "no") are called [[logical matrix|logical matrices]]. The [[distance matrix|distance (or cost) matrix]] contains information about distances of the edges.<ref>{{Harvard citations | last1=Punnen| year=2002|nb = yes}}</ref> These concepts can be applied to [[website]]s connected [[hyperlink]]s or cities connected by roads etc., in which case (unless the road network is extremely dense) the matrices tend to be [[sparse matrix|sparse]], i.e., contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in [[network theory]].

===විශ්ලේෂණය සහ ජ්‍යාමිතිය===
The [[Hessian matrix]] of a [[differentiable function]] ''ƒ'': '''R'''<sup>''n''</sup> → '''R''' consists of the [[second derivative]]s of ''ƒ'' with respect to the several coordinate directions, i.e.<ref>{{Harvard citations|last1 =  Lang|year = 1987a|nb = yes|loc=Ch. XVI.6}}</ref>
:<math>H(f) = \left [\frac {\partial^2f}{\partial x_i \, \partial x_j} \right ].</math>
[[Image:Saddle point.png|left|thumb|At the [[saddle point]] (''x''&nbsp;=&nbsp;0, ''y''&nbsp;=&nbsp;0) (red) of the function ''f''(''x'',&minus;''y'') =&nbsp;''x''<sup>2</sup>&nbsp;&minus;&nbsp;''y''<sup>2</sup>, the Hessian matrix <math>\begin{bmatrix}
2 & 0 \\
0 & -2
\end{bmatrix}</math> is [[indefinite matrix|indefinite]].]]It encodes information about the local growth behaviour of the function: given a [[critical point (mathematics)|critical point]] '''x'''&nbsp;=&nbsp;(''x''<sub>1</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>), i.e., a point where the first [[partial derivatives]] <math>\partial f / \partial x_i</math> of ''ƒ'' vanish, the function has a [[local minimum]] if the Hessian matrix is [[positive definite matrix|positive definite]]. [[Quadratic programming]] can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see [[#quadratic forms|above]]).<ref>{{Harvard citations | last1=Nocedal | year=2006 | loc=Ch. 16|nb=yes}}</ref>

Another matrix frequently used in geometrical situations is the <cite id=Jacobi_matrix>[[Jacobian matrix and determinant|Jacobi matrix]]</cite> of a differentiable map ''f'': '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup>. If ''f''<sub>1</sub>, ..., ''f''<sub>''m''</sub> denote the components of ''f'', then the Jacobi matrix is defined as <ref>{{Harvard citations|last1 =  Lang|year = 1987a|nb = yes|loc=Ch. XVI.1}}</ref>
:<math>J(f) = \left [\frac {\partial f_i}{\partial x_j} \right ]_{1 \leq i \leq m, 1 \leq j \leq n}.</math>
If ''n'' > ''m'', and if the rank of the Jacobi matrix attains its maximal value ''m'', ''f'' is locally invertible at that point, by the [[implicit function theorem]].<ref>{{Harvard citations|last1 =  Lang|year = 1987a|nb = yes|loc=Ch. XVI.5}}. For a more advanced, and more general statement see {{Harvard citations|last1 =  Lang|year = 1969|nb = yes|loc=Ch. VI.2}}</ref>

[[Partial differential equation]]s can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation. For [[elliptic partial differential equation]]s this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question.<ref>{{Harvard citations|last1 =  Gilbarg|last2=Trudinger|year=2001|nb = yes}}</ref>

The [[finite element method]] is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.<ref>{{Harvard citations|last1 =  Šolin|year=2005|nb = yes|loc=Ch. 2.5}}. See also [[stiffness method]].</ref>

{{clear}}

===සම්භාවිතා සිද්ධාන්ත සහ සංඛ්‍යානය===
[[File:Markov chain.png|right|thumb|280px|Two different Markov chains. The chart depicts the number of particles (of a total of 1000) in state "2". Both limiting values can be determined from the transition matrices, which are given by <math>\begin{bmatrix}.7&0\\.3&1\end{bmatrix} </math> (red) and <math>\begin{bmatrix}.7&.2\\.3&.8\end{bmatrix} </math> (black).]]
[[Stochastic matrix|Stochastic matrices]] are square matrices whose rows are [[probability vector]]s, i.e., whose entries sum up to one. Stochastic matrices are used to define [[Markov chain]]s with finitely many states.<ref>{{Harvard citations|last1 =  Latouche |last2=Ramaswami |year = 1999|nb = yes}}</ref> A row of the stochastic matrix gives the probability distribution for the next position of some particle currently in the state that corresponds to the row. Properties of the Markov chain like [[absorbing state]]s, i.e., states that any particle attains eventually, can be read off the eigenvectors of the transition matrices.<ref>{{Harvard citations|last1 =  Mehata|last2=Srinivasan|year = 1978|nb = yes|loc=Ch. 2.8}}</ref>

Statistics also makes use of matrices in many different forms.<ref>{{Citation|last=Healy|first=Michael|title=Matrices for Statistics|year=1986|publisher=[[Oxford University Press]]|isbn=9780198507024|authorlink=Michael Healy (statistician)}}</ref> [[Descriptive statistics]] is concerned with describing data sets, which can often be represented in matrix form, by reducing the amount of data. The [[covariance matrix]] encodes the mutual [[variance]] of several [[random variable]]s.<ref>{{Harvard citations | last1=Krzanowski | year=1988 | loc=Ch. 2.2., p. 60|nb=yes}}</ref> Another technique using matrices are [[linear least squares]], a method that approximates a finite set of pairs (''x''<sub>1</sub>, ''y''<sub>1</sub>), (''x''<sub>2</sub>, ''y''<sub>2</sub>), ..., (''x''<sub>''N''</sub>, ''y''<sub>''N''</sub>), by a linear function
:''y''<sub>''i''</sub> ≈ ''ax''<sub>''i''</sub> + ''b'', ''i'' = 1, ..., ''N''
which can be formulated in terms of matrices, related to the [[singular value decomposition]] of matrices.<ref>{{Harvard citations | last1=Krzanowski | year=1988 | loc=Ch. 4.1|nb=yes}}</ref>

[[Random matrix|Random matrices]] are matrices whose entries are random numbers, subject to suitable [[probability distribution]]s, such as [[matrix normal distribution]]. Beyond probability theory, they are applied in domains ranging from [[number theory]] to [[physics]].<ref>{{Harvard citations | last1=Conrey |year=2007|nb=yes}}</ref><ref>{{Harvard citations | last1=Zabrodin | last2=Brezin  | last3=Kazakov  | last4=Serban | last5=Wiegmann | year=2006|nb=yes}}</ref>

===සමමිති සහ භෞතික විද්‍යාව තුළ පරිණාමණ===
{{See|Symmetry in physics}}
Linear transformations and the associated [[symmetry|symmetries]] play a key role in modern physics. For example, [[elementary particle]]s in [[quantum field theory]] are classified as representations of the [[Lorentz group]] of special relativity and, more specifically, by their behavior under the [[spin group]].  Concrete representations involving the [[Pauli matrices]] and more general [[gamma matrices]] are an integral part of the physical description of [[fermion]]s, which behave as [[spinor]]s.<ref>
{{Harvard citations|last1 =Itzykson|last2=Zuber|year =1980 |nb = yes|loc=Ch. 2}}</ref> For the three lightest [[quark]]s, there is a group-theoretical representation involving the [[special unitary group]] SU(3); for their calculations, physicists use a convenient matrix representation known as the [[Gell-Mann matrices]], which are also used for the SU(3) [[gauge group]] that forms the basis of the modern description of strong nuclear interactions, [[quantum chromodynamics]]. The [[Cabibbo&ndash;Kobayashi&ndash;Maskawa matrix]], in turn, expresses the fact that the basic quark states that are important for [[weak interaction]]s are not the same as, but linearly related to the basic quark states that define particles with specific and distinct [[mass]]es.<ref>
see {{Harvard citations|last1 =Burgess|last2=Moore|year = 2007|nb = yes|loc=section 1.6.3. (SU(3)), section 2.4.3.2. (Kobayashi&ndash;Maskawa matrix)}}</ref>

===ක්වොන්ටම් ගති ස්වභාවයේ රේඛීය සංයෝජන===
The first model of [[quantum mechanics]] ([[Werner Heisenberg|Heisenberg]], 1925) represented the theory's operators by infinite-dimensional matrices acting on quantum states.<ref>{{Harvard citations|last1 =Schiff|year = 1968|nb = yes|loc=Ch. 6}}</ref> This is also referred to as [[matrix mechanics]]. One particular example is the [[density matrix]] that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" [[eigenstates]].<ref>{{Harvard citations|last1 =Bohm|year = 2001|nb = yes|loc=sections II.4 and II.8}}</ref>

Another matrix serves as a key tool for describing the scattering experiments that form the cornerstone of experimental particle physics: Collision reactions such as occur in [[particle accelerator]]s, where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states.  The linear combination is given by a matrix known as the [[S-matrix]], which encodes all information about the possible interactions between particles.<ref>{{Harvard citations|last1 =Weinberg|year =1995|nb = yes|loc=Ch. 3}}</ref>

===සාමාන්‍ය ක්‍රම===
A general application of matrices in physics is to the description of linearly coupled harmonic systems. The [[equation of motion|equations of motion]] of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement vector to characterize the interactions. The best way to obtain solutions is to determine the system's [[eigenvector]]s, its [[normal mode]]s, by diagonalizing the matrix equation. Techniques like this are crucial when it comes to the internal dynamics of [[molecules]]: the internal vibrations of systems consisting of mutually bound component atoms.<ref>{{Harvard citations|last1 =Wherrett|year = 1987|nb = yes|loc=part II}}</ref> They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.<ref>{{Harvard citations|last1 =Riley|last2=Hobson|last3=Bence|year = 1997|nb = yes|loc=7.17}}</ref>

===ජ්‍යාමිතික දෘශ්ටි විද්‍යාව===
[[Geometrical optics]] provides further matrix applications. In this approximative theory, the [[light wave|wave nature]] of light is neglected. The result is a model in which [[light ray]]s are indeed [[ray (geometry)|geometrical rays]]. If the deflection of light rays by optical elements is small, the action of a [[lens (optics)|lens]] or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called [[ray transfer matrix]]: the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. Actually, there are two kinds of matrices, viz. a ''refraction matrix'' describing the refraction at a lens surface, and a ''translation matrix'', describing the translation of the plane of reference to the next refracting surface, where another refraction matrix applies.
The optical system, consisting of a combination of lenses and/or reflective elements, is simply described by the matrix resulting from the product of the components' matrices.<ref>{{Harvard citations|last1 =Guenther|year =1990 |nb = yes|loc=Ch. 5}}</ref>

===ඉලෙක්ට්‍රෝනික විද්‍යාව===
Traditional [[mesh analysis]] in electronics leads to a system of linear equations that can be described with a matrix.

The behaviour of many [[Electronics|electronic]] components can be described using matrices. Let ''A'' be a 2-dimensional vector with the component's input voltage ''v''<sub>1</sub> and input current ''i''<sub>1</sub> as its elements, and let ''B'' be a 2-dimensional vector with the component's output voltage ''v''<sub>2</sub> and output current ''i''<sub>2</sub> as its elements. Then the behaviour of the electronic component can be described by  ''B'' = ''H'' '''·''' ''A'', where ''H'' is a 2 x 2 matrix containing one [[Electrical impedance|impedance]] element (''h''<sub>12</sub>), one [[admittance]] element (''h''<sub>21</sub>) and two [[Dimensionless quantity|dimensionless]] elements (''h''<sub>11</sub> and ''h''<sub>22</sub>). Calculating a circuit now reduces to multiplying matrices.

==ඉතිහාසය==
Matrices have a long history of application in solving [[linear equations]]. The [[Chinese mathematics|Chinese text]] ''[[The Nine Chapters on the Mathematical Art]]'' (''Jiu Zhang Suan Shu''), from between 300 BC and AD 200, is the first example of the use of matrix methods to solve [[System of linear equations|simultaneous equations]],<ref>{{Harvard citations | last1=Shen | last2=Crossley | last3=Lun | year=1999|nb=yes}} cited by {{Harvard citations | last1=Bretscher | year=2005|nb=yes|loc=p. 1}}</ref> including the concept of [[determinant]]s, over 1000 years before its publication by the [[Japanese mathematics|Japanese mathematician]] [[Seki Kowa|Seki]] in 1683{{Citation needed|date=September 2010}} and the German mathematician [[Gottfried Leibniz|Leibniz]] in 1693. [[Gabriel Cramer|Cramer]] presented [[Cramer's rule|his rule]] in 1750.

Early matrix theory emphasized determinants more strongly than matrices and an independent matrix concept akin to the modern notion emerged only in 1858, with [[Arthur Cayley|Cayley's]] ''Memoir on the theory of matrices''.<ref>{{Harvard citations | last1=Cayley  | year=1889|nb=yes|loc=vol. II, p. 475–496}}</ref><ref>{{Harvard citations | editor1-last=Dieudonné| year=1978|loc=Vol. 1, Ch. III, p. 96|nb=yes}}</ref> The term "matrix" ([[Latin]] for "womb", derived from ''[[wikt:mater#Latin|mater]]''&mdash;mother<ref>{{Citation |url=http://www.merriam-webster.com/dictionary/matrix|title=Merriam–Webster dictionary|accessdate=April, 20th 2009 |publisher=[[Merriam–Webster]]}}</ref>) was coined by [[J. J. Sylvester|Sylvester]], who understood a matrix as an object giving rise to a number of determinants today called [[minor (linear algebra)|minor]]s, that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows.<ref>Per the [[Oxford English Dictionary|OED]] the first usage of the word "matrix" with respect to mathematics appears in J. J. Sylvester in London, Edinb. & Dublin Philos. Mag. 37 (1850), p. 369:  "We ‥commence‥ with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order.</ref> In a 1851 paper, Sylvester explains:  
: I have in previous papers defined a "Matrix" as a rectangular array of terms, out of which different systems of determinants may be engendered as from the womb of a common parent.<ref>The Collected Mathematical Papers of James Joseph Sylvester: 1837–1853, [http://books.google.com/books?id=5GQPlxWrDiEC&pg=PA247&dq=sylvester+matrix+womb&hl=en&ei=uJakTaytCoOv8gPa5cG5Dw&sa=X&oi=book_result&ct=result&resnum=8&ved=0CE8Q6AEwBw#v=onepage&q&f=false Paper 37], p. 247</ref>

The study of determinants sprang from several sources.<ref>{{Harvard citations | last1=Knobloch | year=1994 |nb=yes}}</ref> [[number theory|Number-theoretical]] problems led Gauss to relate coefficients of [[quadratic form]]s, i.e., expressions such as {{nowrap|''x''<sup>2</sup> + ''xy'' − 2''y''<sup>2</sup>,}} and [[linear map]]s in three dimensions to matrices. [[Gotthold Eisenstein|Eisenstein]] further developed these notions, including the remark that, in modern parlance, [[matrix product]]s are [[non-commutative]]. [[Augustin Cauchy|Cauchy]] was the first to prove general statements about determinants, using as definition of the determinant of a matrix '''A''' = [''a''<sub>''i'',''j''</sub>] the following: replace the powers ''a''<sub>''j''</sub><sup>''k''</sup> by ''a''<sub>''jk''</sub> in the [[polynomial]]
:<math>a_1 a_2 \cdots a_n \prod_{i < j} (a_j - a_i)\;,</math>
where Π denotes the [[multiplication|product]] of the indicated terms. He also showed, in 1829, that the [[eigenvalue]]s of symmetric matrices are real.<ref>{{Harvard citations | last1=Hawkins | year=1975 |nb=yes}}</ref> [[Carl Gustav Jakob Jacobi|Jacobi]] studied "functional determinants"—later called [[Jacobian matrix and determinant|Jacobi determinant]]s by Sylvester—which can be used to describe geometric transformations at a local (or [[infinitesimal]]) level, see [[#Jacobi matrix|above]]; [[Leopold Kronecker|Kronecker's]] ''Vorlesungen über die Theorie der Determinanten''<ref>{{Harvard citations | last1=Kronecker | editor1-last=Hensel |year=1897|nb=yes}}</ref> and [[Karl Weierstrass|Weierstrass']] ''Zur Determinantentheorie'',<ref>{{Harvard citations | last1=Weierstrass | year=1915 |volume=3|loc=pp. 271–286|nb=yes}}</ref> both published in 1903, first treated determinants [[axiom]]atically, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy. At that point, determinants were firmly established.

Many theorems were first established for small matrices only, for example the [[Cayley–Hamilton theorem]] was proved for 2×2 matrices by Cayley in the aforementioned memoir, and by [[William Rowan Hamilton|Hamilton]] for 4×4 matrices. [[Georg Frobenius|Frobenius]], working on [[bilinear form]]s, generalized the theorem to all dimensions (1898). Also at the end of the 19th century the [[Gauss–Jordan elimination]] (generalizing a special case now known as [[Gauss elimination]]) was established by [[Wilhelm Jordan (geodesist)|Jordan]]. In the early 20th century, matrices attained a central role in linear algebra.<ref>{{Harvard citations | last1=Bôcher | year=2004|nb=yes}}</ref> partially due to their use in classification of the [[hypercomplex number]] systems of the previous century.

The inception of [[matrix mechanics]] by [[Werner Heisenberg|Heisenberg]], [[Max Born|Born]] and [[Pascual Jordan|Jordan]] led to studying matrices with infinitely many rows and columns.<ref>{{Harvard citations | last1=Mehra | last2=Rechenberg | year=1987|nb=yes}}</ref> Later, [[John von Neumann|von Neumann]] carried out the [[mathematical formulation of quantum mechanics]], by further developing [[functional analysis|functional analytic]] notions such as [[linear operator]]s on [[Hilbert space]]s, which, very roughly speaking, correspond to [[Euclidean space]], but with an infinity of [[Hamel dimension|independent directions]].

===ලෝක ඉතිහාසයේ ගණිතය තුළ න්‍යසයන්හි වෙනත් භාවිත===
The word has been used in unusual ways by at least two authors of historical importance.

[[Bertrand Russell]] and [[Alfred North Whitehead]] in their ''Principia Mathematica'' (1910–1913) use the word '''matrix''' in the context of their [[Axiom of reducibility]]. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the “bottom” (0 order) the function is identical to its [[Extension (predicate logic)|extension]]:
:“Let us give the name of ''matrix'' to any function, of however many variables, which does not involve any [[apparent variable]]s. Then any possible function other than a matrix is derived from a matrix by means of generalization, i.e., by considering the proposition which asserts that the function in question is true with all possible values or with some value of one of the arguments, the other argument or arguments remaining undetermined”.<ref>Alfred North Whitehead and Bertrand Russell (1913) ''Principia Mathematica to *56'', Cambridge at the University Press, Cambridge UK (republished 1962) cf page 162ff.</ref>
For example a function Φ(''x, y'') of two variables ''x'' and ''y'' can be reduced to a ''collection'' of functions of a single variable, e.g., ''y'', by “considering” the function for all possible values of “individuals” ''a<sub>i</sub>'' substituted in place of variable ''x''. And then the resulting collection of functions of the single variable ''y'', i.e., ∀a<sub>i</sub>: Φ(''a<sub>i</sub>, y''), can be reduced to a “matrix” of values by “considering” the function for all possible values of “individuals” ''b<sub>i</sub>'' substituted in place of variable ''y'':
:∀b<sub>j</sub>∀a<sub>i</sub>: Φ(''a<sub>i</sub>, ''b<sub>j</sub>'').

[[Alfred Tarski]] in his 1946 ''Introduction to Logic'' used the word “matrix” synonymously with the notion of [[truth table]] as used in mathematical logic.<ref>Tarski, Alfred 1946 ''Introduction to Logic and the Methodology of Deductive Sciences'', Dover Publications, Inc, New York NY, ISBN 0-486-28462-X.</ref>

==See also==
{{Portal|Mathematics}}
{{Div col|cols=2}}

* [[Algebraic multiplicity]]
* [[Geometric multiplicity]]
* [[Gram-Schmidt process]]
* [[List of matrices]]
* [[Matrix calculus]]
* [[Periodic matrix set]]
* [[Tensor]]

{{Div col end}}

==සටහන්==
{{Reflist|colwidth=30em}}
{{Reflist|group=nb|3}}

==නිර්දේශ==
* {{Citation | last1=Arnold | first1=V. I. | author1-link=Vladimir Arnold | last2=Cooke | first2=Roger | author2-link=Roger Cooke | title=Ordinary differential equations | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-54813-3 | year=1992}}
* {{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | title=Algebra | publisher=[[Prentice Hall]] | isbn=978-0-89871-510-1 | year=1991}}
* {{Citation | last1=Association for Computing Machinery | title=Computer Graphics | publisher=Tata McGraw–Hill | isbn=978-0-07-059376-3 | year=1979}}
* {{Citation | last1=Baker | first1=Andrew J. | title=Matrix Groups: An Introduction to Lie Group Theory | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-1-85233-470-3 | year=2003}}
* {{Citation | last1=Bau III | first1=David | last2=Trefethen | first2=Lloyd N. | author2-link=Lloyd Nicholas Trefethen | title=Numerical linear algebra | publisher=Society for Industrial and Applied Mathematics | location=Philadelphia | isbn=978-0-89871-361-9 | year=1997}}
* {{Citation | last1=Bretscher | first1=Otto | title=Linear Algebra with Applications | publisher=Prentice Hall | edition=3rd | year=2005}}
* {{Citation | last1=Bronson | first1=Richard | title=Schaum's outline of theory and problems of matrix operations | publisher=[[McGraw–Hill]] | location=New York | isbn=978-0-07-007978-6 | year=1989}}
* {{Citation | last1=Brown | first1=William A. | title=Matrices and vector spaces | publisher=M. Dekker | location=New York | isbn=978-0-8247-8419-5 | year=1991}}
* {{Citation | last1=Coburn | first1=Nathaniel | title=Vector and tensor analysis | publisher=Macmillan | location=New York | oclc=1029828 | year=1955}}
* {{Citation | last1=Conrey | first1=J. B. | title=Ranks of elliptic curves and random matrix theory | publisher=[[Cambridge University Press]] | isbn=978-0-521-69964-8 | year=2007}}
* {{Citation | last1=Fudenberg | first1=D. | last2=Tirole | first2=Jean | author2-link=Jean Tirole | title=Game Theory | publisher=[[MIT Press]] | year=1983}}
* {{Citation | author2-link=Neil Trudinger | last1=Gilbarg | first1=David | last2=Trudinger | first2=Neil S. | title=Elliptic partial differential equations of second order | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | isbn=978-3-540-41160-4 | year=2001}}
* {{Citation | first1=Chris|last1=Godsil|authorlink1=Chris Godsil|first2=Gordon|last2=Royle|authorlink2=Gordon Royle|title=Algebraic Graph Theory| publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-95220-8 | year=2004 | volume=207}}
* {{Citation | last1=Golub | first1=Gene H. | author1-link=Gene H. Golub | last2=Van Loan | first2=Charles F. | author2-link=Charles F. Van Loan | title=Matrix Computations | publisher=Johns Hopkins | edition=3rd | isbn=978-0-8018-5414-9 | year=1996}}
* {{Citation | last1=Greub | first1=Werner Hildbert | title=Linear algebra | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90110-7 | year=1975}}
* {{Citation | last1=Halmos | first1=Paul Richard | title=A Hilbert space problem book | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-90685-0 | mr=675952 | year=1982 | volume=19}}
* {{Citation | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis | publisher=Cambridge University Press | isbn=978-0-521-38632-6 | year=1985}}
* {{Citation | last1=Householder | first1=Alston S. | title=The theory of matrices in numerical analysis | publisher=[[Dover Publications]] | location=New York | mr=0378371 | year=1975}}
* {{Citation | last1=Krzanowski | first1=W. J. | title=Principles of multivariate analysis | publisher=The Clarendon Press Oxford University Press | series=Oxford Statistical Science Series | isbn=978-0-19-852211-9 | mr=969370 | year=1988 | volume=3}}
* {{Citation | editor1-last=Itõ | editor1-first=Kiyosi | title=Encyclopedic dictionary of mathematics. Vol. I--IV | publisher=MIT Press | edition=2nd | isbn=978-0-262-09026-1 | mr=901762 | year=1987}}
* {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Analysis II | publisher=[[Addison-Wesley]] | year=1969}}
* {{Citation | last1=Lang | first1=Serge | title=Calculus of several variables | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | isbn=978-0-387-96405-8 | year=1987a}}
* {{Citation | last1=Lang | first1=Serge | title=Linear algebra | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-96412-6 | year=1987b}}
* {{Lang Algebra}}
* {{Citation | last1=Latouche | first1=G. | last2=Ramaswami | first2=V. | title=Introduction to matrix analytic methods in stochastic modeling | publisher=Society for Industrial and Applied Mathematics | location=Philadelphia | edition=1st | isbn=978-0-89871-425-8 | year=1999}}
* {{Citation | last1=Manning | first1=Christopher D. | last2=Schütze | first2=Hinrich | title=Foundations of statistical natural language processing | publisher=MIT Press | isbn=978-0-262-13360-9 | year=1999}}
* {{Citation | last1=Mehata | first1=K. M. | last2=Srinivasan | first2=S. K. | title=Stochastic processes | publisher=McGraw–Hill | location=New York | isbn=978-0-07-096612-3 | year=1978}}
* {{Citation | last1=Mirsky | first1=Leonid | title=An Introduction to Linear Algebra | url=http://books.google.com/?id=ULMmheb26ZcC&pg=PA1&dq=linear+algebra+determinant | publisher=Courier Dover Publications | isbn=978-0-486-66434-7 | year=1990}}
* {{Citation | last1=Nocedal | first1=Jorge | last2=Wright | first2=Stephen J. | title=Numerical Optimization | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | isbn=978-0-387-30303-1 | year=2006 | page=449 }}
* {{Citation | last1=Oualline | first1=Steve | title=Practical C++ programming | publisher=[[O'Reilly Media|O'Reilly]] | isbn=978-0-596-00419-4 | year=2003}}
* {{Citation | last1=Press | first1=William H. | last2=Flannery | first2=Brian P. | last3=Teukolsky | first3=Saul A. | author3-link=Saul Teukolsky | last4=Vetterling | first4=William T. | title=Numerical Recipes in FORTRAN: The Art of Scientific Computing | url=http://www.mpi-hd.mpg.de/astrophysik/HEA/internal/Numerical_Recipes/f2-3.pdf | publisher=Cambridge University Press | edition=2nd | year=1992 | chapter=LU Decomposition and Its Applications | pages=34–42}}
* {{Citation | last1=Punnen | first1=Abraham P. | last2=Gutin | first2=Gregory | title=The traveling salesman problem and its variations | publisher=Kluwer Academic Publishers | location=Boston | isbn=978-1-4020-0664-7 | year=2002}}
* {{Citation | last1=Reichl | first1=Linda E. | title=The transition to chaos: conservative classical systems and quantum manifestations | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-98788-0 | year=2004}}
* {{Citation | last1=Rowen | first1=Louis Halle | title=Graduate Algebra: noncommutative view | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-4153-2 | year=2008}}
* {{Citation | last1=Šolin | first1=Pavel | title=Partial Differential Equations and the Finite Element Method | publisher=[[Wiley-Interscience]] | isbn=978-0-471-76409-0 | year=2005}}
* {{Citation | last1=Stinson | first1=Douglas R. | title=Cryptography | publisher=Chapman & Hall/CRC | series=Discrete Mathematics and Its Applications | isbn=978-1-58488-508-5 | year=2005}}
* {{Citation | last1=Stoer | first1=Josef | last2=Bulirsch | first2=Roland | title=Introduction to Numerical Analysis | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | isbn=978-0-387-95452-3 | year=2002}}
* {{Citation | last1=Ward | first1=J. P. | title=Quaternions and Cayley numbers | publisher=Kluwer Academic Publishers Group | location=Dordrecht | series=Mathematics and its Applications | isbn=978-0-7923-4513-8 | mr=1458894 | year=1997 | volume=403}}
* {{Citation | last1=Wolfram | first1=Stephen | author1-link=Stephen Wolfram | title=The Mathematica Book | publisher=Wolfram Media | location=Champaign, Ill | edition=5th | isbn=978-1-57955-022-6 | year=2003}}

===භෞතික විද්‍යා නිර්දේශ===
* {{Citation|last=Bohm|first=Arno|title=Quantum Mechanics: Foundations and Applications|publisher=Springer|year=2001|isbn=0-387-95330-2}}
* {{Citation|last=Burgess|first=Cliff|last2=Moore|first2=Guy|title=The Standard Model. A Primer|publisher=Cambridge University Press|year=2007|isbn=0-521-86036-9}}
*{{Citation|last=Guenther|first=Robert D.|title=Modern Optics|publisher=John Wiley|year=1990|isbn=0-471-60538-7}}
*{{Citation|last=Itzykson|first=Claude|last2=Zuber|first2=Jean-Bernard|title=Quantum Field Theory|publisher=McGraw–Hill|year=1980|isbn=0-07-032071-3}}
*{{Citation|last=Riley|first=K. F.|last2=Hobson|first2=M. P.|last3=Bence|first3=S. J.|title=Mathematical methods for physics and engineering|publisher=Cambridge University Press|year=1997|isbn=0-521-55506-X}}
*{{Citation|last=Schiff|first=Leonard I.|title=Quantum Mechanics|edition=3rd|publisher=McGraw–Hill|year=1968}}
*{{Citation|last=Weinberg|first=Steven|title=The Quantum Theory of Fields. Volume I: Foundations|publisher=Cambridge University Press|year=1995|isbn=0-521-55001-7}}
* {{Citation|last=Wherrett|first=Brian S.|year=1987|title=Group Theory for Atoms, Molecules and Solids|publisher=Prentice–Hall International|isbn=0-13-365461-3}}
* {{Citation | last1=Zabrodin | first1=Anton | last2=Brezin | first2=Édouard | last3=Kazakov | first3=Vladimir | last4=Serban | first4=Didina | last5=Wiegmann | first5=Paul | title=Applications of Random Matrices in Physics (NATO Science Series II: Mathematics, Physics and Chemistry) | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-1-4020-4530-1 | year=2006}}

===පෞරාණික නිර්දේශ===
* {{Citation | last1=Bôcher | first1=Maxime | author1-link=Maxime Bôcher | title=Introduction to higher algebra | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-49570-5 | year=2004}}, reprint of the 1907 original edition
* {{Citation | last1=Cayley | first1=Arthur | author1-link=Arthur Cayley | title=The collected mathematical papers of Arthur Cayley | url=http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=ABS3153.0001.001;didno=ABS3153.0001.001;view=image;seq=00000140 | publisher=[[Cambridge University Press]] | year=1889 | volume=I (1841–1853) | pages=123–126}}
* {{Citation | editor1-last=Dieudonné | editor1-first=Jean | editor1-link=Jean Dieudonné | title=Abrégé d'histoire des mathématiques 1700-1900 | publisher=Hermann | location=Paris | year=1978}}
* {{Citation | last1=Hawkins | first1=Thomas | title=Cauchy and the spectral theory of matrices | mr=0469635 | year=1975 | journal=[[Historia Mathematica]] | issn=0315-0860 | volume=2 | pages=1–29 | doi=10.1016/0315-0860(75)90032-4}}
* {{Citation | last1=Knobloch | first1=Eberhard | title=The intersection of history and mathematics | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Sci. Networks Hist. Stud. | mr=1308079 | year=1994 | volume=15 | chapter=From Gauss to Weierstrass: determinant theory and its historical evaluations | pages=51–66}}
* {{Citation | last1=Kronecker | first1=Leopold | author1-link=Leopold Kronecker | editor1-last=Hensel | editor1-first=Kurt | editor1-link=Kurt Hensel | title=Leopold Kronecker's Werke | url=http://name.umdl.umich.edu/AAS8260.0002.001 | publisher=Teubner | year=1897}}
* {{Citation | last1=Mehra | first1=J. | last2=Rechenberg | first2=Helmut | title=The Historical Development of Quantum Theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=1st | isbn=978-0-387-96284-9 | year=1987}}
* {{Citation | last1=Shen | first1=Kangshen | last2=Crossley | first2=John N. | last3=Lun | first3=Anthony Wah-Cheung | title=Nine Chapters of the Mathematical Art, Companion and Commentary | publisher=[[Oxford University Press]] | edition=2nd | isbn=978-0-19-853936-0 | year=1999}}
* {{Citation | last1=Weierstrass | first1=Karl | author1-link=Karl Weierstrass | title=Collected works | url=http://name.umdl.umich.edu/AAN8481.0003.001 | year=1915 | volume=3}}

==භාහිර ඈඳුම්==
{{Wikibooks|Linear Algebra|Matrices}}
{{Wikiversity|at=Linear algebra#Matrices|Matrices}}
; History
*[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html MacTutor: Matrices and determinants]
*[http://www.economics.soton.ac.uk/staff/aldrich/matrices.htm Matrices and Linear Algebra on the Earliest Uses Pages]
*[http://jeff560.tripod.com/matrices.html Earliest Uses of Symbols for Matrices and Vectors]

; Online books
* {{Citation | last1=Kaw | first1=Autar K. | title=Introduction to Matrix Algebra | url=http://autarkaw.com/books/matrixalgebra/index.html | isbn=978-0-615-25126-4}}
* {{Citation | title= The Matrix Cookbook | url=http://matrixcookbook.com | accessdate=12/10/2008}}
* {{Citation | first1=M. | last1=Brookes | title=The Matrix Reference Manual | url=http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html | publisher=[[Imperial College]] | location=London|year=2005|accessdate=12/10/2008}}

; Online matrix calculators
* {{Citation|title=SuperiorMath (Matrix Calculator)|url=http://www.webalice.it/simoalessia/SuperiorMath/matrix.html|access-date=2011-11-29|archive-date=2011-09-28|archive-url=https://web.archive.org/web/20110928084531/http://www.webalice.it/simoalessia/SuperiorMath/matrix.html}}
* {{Citation  |title=Matrix Calculator (DotNumerics )| url=http://www.dotnumerics.com/MatrixCalculator/ }}
* {{Citation | last1=Xiao |first1=Gang |title= Matrix calculator| url=http://wims.unice.fr/wims/wims.cgi?module=tool/linear/matrix.en| accessdate=12/10/2008}}
* {{Citation|title=Online matrix calculator|url=http://www.bluebit.gr/matrix-calculator/|accessdate=12/10/2008|access-date=2011-11-29|archive-date=2008-12-12|archive-url=https://web.archive.org/web/20081212221215/http://www.bluebit.gr/matrix-calculator/}}
* {{Citation|title=Online matrix calculator(ZK framework)|url=http://matrixcalc.info/MatrixZK/|accessdate=11/26/2009|access-date=2011-11-29|archive-date=2013-05-12|archive-url=https://web.archive.org/web/20130512101418/http://matrixcalc.info/MatrixZK/}}

* {{Citation| title=MacAnova|url=http://www.stat.umn.edu/macanova/macanova.home.html|last1=Oehlert|first1=Gary W. |last2=Bingham|first2= Christopher |publisher=[[University of Minnesota]], School of Statistics|accessdate=12/10/2008}}, a freeware package for matrix algebra and statistics
* {{Citation  |title=Online matrix calculator| url=http://www.idomaths.com/matrix.php | accessdate=12/14/2009}}
*[http://www.stud.feec.vutbr.cz/~xvapen02/vypocty/matreg.php?language=english Operation with matrices in R (determinant, track, inverse, adjoint, transpose)] {{Webarchive|url=https://web.archive.org/web/20110718192056/http://www.stud.feec.vutbr.cz/~xvapen02/vypocty/matreg.php?language=english |date=2011-07-18 }}



{{DEFAULTSORT:Matrix (Mathematics)}}
[[Category:Abstract algebra]]
[[Category:Linear algebra]]
[[Category:Matrices| ]]






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