matrix

A matrix is simply a mapping $M\\colon A\\times B\\to C$ of the product of twosets into some third set.As a rule, though, the word matrix and the notation associated with itare used only in connection with linear mappings.In such cases $C$ is the ring or field of scalars.

Matrix of a linear mapping

Definition: Let $V$ and $W$ be finite-dimensional vectorspaces over the same field $k$ , with bases $A$ and $B$ respectively,and let $f\\colon V\\to W$ be a linear mapping.For each $a\\in A$ let $(k_{ab})_{b\\in B}$ be the unique family of scalars(elements of $k$ ) such that

f(a)=\\sum_{b\\in B}k_{ab}b\\;.

Then the family $(M_{ab})$ (or equivalently the mapping $(a,b)\\mapsto M_{ab}$ of $A\\times B\\to k$ )is called the matrix of $f$ with respect to the given bases $A$ and $B$ .The scalars $M_{ab}$ are called the components of the matrix. The matrix $M$ is said to be of size $\\lvert A\\rvert$ -by- $\\lvert B\\rvert$ or simply $\\lvert A\\rvert$ x $\\lvert B\\rvert$ matrix.

The matrix describes the function $f$ completely; for any element

x=\\sum_{a\\in A}x_{a}a

of $V$ , we have

f(x)=\\sum_{a\\in A}M_{ab}b

as is readily verified.

Any two linear mappings $V\\to W$ have a sum, defined pointwise; itis easy to verify that the matrix of the sum is the sum, componentwise,of the two given matrices.

The formalism of matrices extends somewhat to linear mappings betweenmodules, i.e. extends to a ring $k$ , not necessarily commutative,rather than just a field.

Rows and columns; product of two matrices

Suppose we are given three modules $V, W, X$ , with bases $A, B, C$ respectively,and two linear mappings $f\\colon V\\to W$ and $g\\colon W\\to X$ . $f$ and $g$ have some matrices $(M_{ab})$ and $(N_{bc})$ with respect tothose bases. The product matrix $N M$ is defined as the matrix $(P_{ac})$ of the function

x\\mapsto g(f(x))

V\\to W

with respect to the bases $A$ and $C$ . Straight from the definitionsof a linear mapping and a basis, one verifies that

P_{ac}=\\sum_{b\\in B}M_{ab}N_{bc}

(1)

for all $a\\in A$ and $c\\in C$ .

To illustrate the notation of matrices in terms of rows and columns,suppose the spaces $V, W, X$ have dimensions 2, 3, and 2 respectively, and bases

A=\\{a_{1},a_{2}\\}\\qquad B=\\{b_{1},b_{2},b_{3}\\}\\qquad C=\\{c_{1},c_{2}\\}\\;.

We write

\\begin{pmatrix}M_{11}&M_{12}&M_{13}\\\\M_{21}&M_{22}&M_{23}\\end{pmatrix}\\begin{pmatrix}N_{11}&N_{12}\\\_{21}&N_{22}\\\_{31}&N_{32}\\end{pmatrix}=\\begin{pmatrix}P_{11}&P_{12}\\\\P_{21}&P_{22}\\end{pmatrix}\\;.

(Notice that we have taken a liberty with the notation,by writing e.g. $M_{12}$ instead of $M_{a_{1}a_{2}}$ .)The equation (1) shows that the multiplicationof two matrices proceeds “rows by columns”. Also, in an expression suchas $N_{23}$ , the first index refers to the row, and the second to thecolumn, in which that component appears.

Similar notation can describe the calculation of $f(x)$ whenever $f$ is a linear mapping. For example, if $f\\colon V\\to W$ is linear,and $x=\\sum_{i}x_{i}a_{i}$ and $f(x)=\\sum_{i}y_{i}b_{i}$ , we write

\\begin{pmatrix}x_{1}&x_{2}\\end{pmatrix}\\begin{pmatrix}M_{11}&M_{12}&M_{13}\\\\M_{21}&M_{22}&M_{23}\\end{pmatrix}=\\begin{pmatrix}y_{1}&y_{2}&y_{3}\\\\\\end{pmatrix}\\;.

When, as above, a “row vector” denotes an element of a space,a “column vector” denotes an element of the dual space.If, say, $\\overline{f}\\colon W^{*}\\to V^{*}$ is the transpose of $f$ ,then, with respect to the bases dual to $A$ and $B$ ,an equation $\\overline{f}(\\sum_{j}\u_{j}\\beta_{j})=\\sum_{i}\\mu_{i}\\alpha_{i}$ may be written

\\begin{pmatrix}\\mu_{1}\\\\\\mu_{2}\\end{pmatrix}=\\begin{pmatrix}M_{11}&M_{12}&M_{13}\\\\M_{21}&M_{22}&M_{23}\\end{pmatrix}\\begin{pmatrix}\u_{1}\\\\\u_{2}\\\\\u_{3}\\end{pmatrix}\\;,

One more illustration: Given a bilinear form $L\\colon V\\times W\\to k$ ,we can denote $L(v,w)$ by

\\begin{pmatrix}v_{1}&v_{2}\\end{pmatrix}\\begin{pmatrix}L_{11}&L_{12}&L_{13}\\\\L_{21}&L_{22}&L_{23}\\end{pmatrix}\\begin{pmatrix}w_{1}\\\\w_{2}\\\\w_{3}\\end{pmatrix}\\;.

Square matrix

A matrix $M\\colon A\\times B\\to C$ is called square if $A=B$ , or if somebijection $A\\to B$ is implicit in the context.(It is not enough for $A$ and $B$ to be equipotent.)Square matrices naturally arise in connection with a linear mapping ofa space into itself (called an endomorphism), and in the relatedcase of a change of basis (from one basis of some space, to anotherbasis of the same space). When $A$ is finite of cardinality $n$ (and thus, so is $B$ ), then $n$ is often called the order of the matrix $M$ . Unfortunately, equally often order of $M$ means the order (http://planetmath.org/OrderGroup) of $M$ as an element of the group $GL_{n}(C)$ (http://planetmath.org/GeneralLinearGroup).

Miscelleous usages of “matrix”

The word matrix has come into use in some areas where linear mappingsare not at issue. An example would be a combinatorical statement,such as Hall’s marriage theorem, phrased in terms of “0-1 matrices”instead of subsets of $A\\times B$ .

Remark

Matrices are heavily used in the physical sciences, engineering,statistics, and computer programming. But for purely mathematical purposes,they are less important than one might expect, and indeed are frequentlyirrelevant in linear algebra. Linear mappings, determinants, traces,transposes, and a number of other simple notions can and should bedefined without matrices, simply because they have a meaning independentof any basis or bases.Many little theorems in linear algebra can be proved in a simplerand more enlightening way without matrices than with them.One more illustration: The derivative (at a point)of a mapping from one surface to another is a linear mapping; it is not amatrix of partial derivatives, because the matrix depends on a choiceof basis but the derivative does not.