Matrix

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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.

Contents

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:

\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}

would equal

\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}

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:

\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} e \\ f \end{bmatrix}=\begin{bmatrix} ae+bf \\ ce+df \end{bmatrix}

It is important to note that matrix multiplication is not commutative: in general, AB \neq BA for two matrices A and B. 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, AB, which is itself a matrix. Then form the product ABx. Matrix multiplication is associative, so we can consider this as either (AB)x or A(Bx). In suffix notation,

∑(AB)ijxj = ∑Aik(Bx)k = ∑AikBkjxj
jkj,k

The vector x is arbitrary, so we can therefore deduce the rule for finding the product of two matrices:

(AB)ij = ∑AikBkj
k

Matrix concepts

Basic concepts

Advanced concepts

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