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Undo revision 710487 by Neoconservative (Talk)
''For the 1999 film, see [[The Matrix]].''
A '''matrix''' (pl.: "matrices," [[Latin]] origin) is a complex ordering, in deliberate fashion, of [[numeral|numerals]]. In [[mathematics]], a "matrix" is a regular grid of numbers, which may be manipulated and solved through intermediate-level [[algebra]]. Matrix algebra is usually taught in [[sophomore]] [[high school]] level mathematics.
More formally, a matrix is an example of a rank-2 [[tensor]].
Alternately, a '''matrix''' may also be a complex ordering of a group of equivalent objects, especially where the order is imposed to gain incidental benefit from the [[synergy]] of the networked objects.
==Mathematics==
In mathematics, matrices can be manipulated in a variety of ways, including [[addition]] and [[multiplication]].
===Addition of matrices===
For example, to add two matrices, one would add their respective elements, thus:
<math>\begin{bmatrix}
x & y & z \\
1 & 3 & 5 \\
0 & 2 & 0
\end{bmatrix} + \begin{bmatrix}
0 & 3 & 1 \\
4 & 3 & {x+2} \\
0 & 4 & v
\end{bmatrix} </math>
would equal
<math>\begin{bmatrix}
{x+0} & {y+3} & {z+1} \\
{1+4} & {3+3} & {5+(x+2)} \\
{0+0} & {2+4} & {0+v}
\end{bmatrix} = \begin{bmatrix}
x & {y+3} & {z+1} \\
5 & 6 & {x+7} \\
0 & 6 & v
\end{bmatrix} </math>
===Multiplication of matrices===
To multiply two matrices, one uses the rule "go along the rows and down the columns". This is best illustrated by a specific example: a matrix times a vector:
<math>\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} \begin{bmatrix}
e \\ f
\end{bmatrix}=\begin{bmatrix}
ae+bf \\
ce+df \end{bmatrix}
</math>
It is important to note that matrix multiplication is not commutative: in general, <math>AB \neq BA</math> for two matrices <math>A</math> and <math>B</math>. This has important consequences in [[quantum mechanics]].
To see why matrix multiplication works the way it does, we will use [[suffix notation]]. Consider first forming the product of two matrices, <math>AB</math>, which is itself a matrix. Then form the product <math>ABx</math>. Matrix multiplication is associative, so we can consider this as either <math>(AB)x</math> or <math>A(Bx)</math>. In suffix notation,
<math>
\sum_j(AB)_{ij}x_j = \sum_k A_{ik}(Bx)_k=\sum_{j,k}A_{ik}B_{kj}x_j
</math>
The vector <math>x</math> is arbitrary, so we can therefore deduce the rule for finding the product of two matrices:
<math>
(AB)_{ij}=\sum_k A_{ik}B_{kj}
</math>
===Matrix concepts===
====Basic concepts====
*[[Adjoint]]
*[[Determinant]]
*[[Diagonal matrix]]
*[[Identity matrix]]
*[[Inverse matrix]]
*[[Null, column and row space]]
*[[Scalar]]
*[[Trace]]
*[[Transpose matrix]]
*[[Vector]]
*[[Zero matrix]]
====Advanced concepts====
*[[Basis]]
*[[Diagonalizeable]]
*[[Eigenspace]]
*[[Eigenvalue]]
*[[Eigenvector]]
*[[Gram-Schmidt process]]
*[[Hermitian matrix]]
*[[Jordan canonical form]]
*[[Laplacian]]
*[[Linear independence]]
*[[Matrix diagonalization]]
*[[Matrix reformation]]
*[[Matrix transformation]]
*[[Orthogonal matrix]]
*[[Orthonormal matrix]]
*[[Resultant]]
*[[Span]]
*[[Systems of linear equations]]
*[[Transcriptor]]
*[[Wronskian]]
[[Category:Linear algebra]]
[[Category:Computers]]