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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.
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.
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==
==Mathematics==
In mathematics, matrices can be manipulated in a variety of ways, including [[addition]] and multiplication. For example, to add to matrices, one would add their respective units, thus:
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}
<math>\begin{bmatrix}
0 & 6 & v
0 & 6 & v
\end{bmatrix} </math>
\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>
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Computers]]
[[Category:Computers]]