330 maximum/minimum

The scalar product and the matrix sum satisfy the

relation:

k(A+ B) = kA + kB

3. Matrix Multiplication: The

DOT PRODUCT

of two vec-

tors provides a natural way to obtain a single numeri-

cal value from two separate lists of numbers: the

product of (x1,x2,…,xn) and (y1,y2,…,yn) is given by

the sum x1y1+ x2y2+…+xnyn. Each row and column

of a matrix provides a list of numbers, and so one can

create from two matrices Aand Ba new array of

numerical values whose entries are the dot products of

the rows or columns of Awith the rows or the

columns of B. It has proved to be convenient to just

use the rows of Aand the columns of B, provided that

the number of entries in each row of Amatches the

number of entries in each column of B. In summary:

If Ais an n×mmatrix and Ban m×rmatrix,

then the matrix product AB is the n×rmatrix

whose (i,j)th entry is the dot product of the ith

row of Awith the jth column of B. We have:

(AB)ij = Ai1B1j+ Ai2B2j+…+AimBmj

For example, if and ,

then AB is the 2 ×2 matrix:

In this example, the product BA is not defined.

In many applications it is assumed that all matrices

are square matrices, that is, they have equal numbers of

rows and columns. In this setting, even though the

products AB and BA of two square matrices Aand B

of the same size may each be defined, they are likely to

be unequal. Thus the matrix product does not satisfy

the

COMMUTATIVE PROPERTY

. The

IDENTITY MATRIX

is

a matrix Iwith the property that AI=IA=Afor any

square matrix Aof a fixed size.

The transpose of a matrix A, denoted AT, is the

matrix obtained from Aby interchanging its rows with

its columns. The product ATB is the matrix whose

(i,j)th entry is the dot product of the ith column of A

with the jth column of B. Similarly, the product ABT

has (i,j)th entry the dot product of the ith row of A

with the jth row of B, and ATBThas (i,j)th entry the dot

product of the ith column of Awith the jth row of B.

This latter example equals the transpose of the original

product BA. We thus have: (BA)T= ATBT.

If one

LINEAR TRANSFORMATION

is represented by

a matrix Aand a second by the matrix B, then the

COMPOSITION

of these two transformations is repre-

sented by a matrix equal to the product BA. (This is

read backwards: the transformation represented by Ais

applied first and is followed by the second transforma-

tion B.) It is precisely the desire to make this observa-

tion hold true that first led mathematicians to define

the matrix product in the manner described above.

See also

DETERMINANT

;

GENERAL LINEAR GROUP

;

INVERSE MATRIX

.

maximum/minimum The highest point on the

graph of a function is called the maximum point of the

graph, and the value of the function at that point is

called the maximum value of the function (or its global

maximum or absolute maximum). Similarly, the mini-

mum point of the graph is the point at which the graph

has its lowest value, and the minimum value of the

graph is the value of the function at that point (also

called the global minimum or absolute minimum). It is

possible for a function to have no maximum value or

no minimum value. For example, the function y= x,

defined over all real numbers, has no maximum or

minimum value, and the function , defined

over all real numbers, has no minimum value. The

EXTREME

-

VALUE THEOREM

shows, on the other hand,

that every continuous function defined on a closed

interval necessarily adopts both a maximum and a min-

imum value on that interval.

A local maximum (also called a relative maxi-

mum) for a function is a point on the graph of a func-

tion that is higher than all its nearby points on the

graph. Clearly, a local maximum need not be the high-

est point on the graph, although the highest point cer-

tainly qualifies as a local maximum. Similarly, a local