Matrix

For the 1999 film, see The Matrix.

A matrix (pl.: "matrices," Latin origin) is a complex ordering, in deliberate fashion, of 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:

Matrix concepts

Basic concepts

Advanced concepts