matrix operations 329

+

have the same magnitude.) The magnitude of a

VECTOR

is its length.

Physicists and astronomers use the phrase

ORDER OF

MAGNITUDE

to refer to the smallest power of 10 needed

to represent a quantity. For example, the numbers 4×1023

and 9.78×1023 are of the same order of magnitude.

See also

SCIENTIFIC NOTATION

.

Mandelbrot set See

FRACTAL

.

matrix (plural, matrices) A rectangular array of

numbers displayed in rows and columns and enclosed

in parentheses is called a matrix. (In science, the word

matrix is used to describe the background material, soil

or rock, that holds an object such as a fossil or a crys-

tal in place. In mathematics, the word is used to

describe an array that “holds” numbers in place.) An m

×nmatrix has mrows and ncolumns. For example,

the object below is a 2 ×3 matrix:

A matrix is called square if the number of rows equals

the number of columns. A matrix with just one row (a

row matrix) or just one column (a column matrix) can

be regarded as a

VECTOR

.

Typically a matrix is denoted with a capital letter.

For instance, the matrix above might be called A,

and the entry in the ith row and jth column as Aij. For

instance, in this example, A11 = 5 and A23 = 2.

Matrices arise in the study of

SIMULTANEOUS LIN

-

EAR EQUATIONS

and the study of

LINEAR ALGEBRA

. Any

LINEAR TRANSFORMATION

can be represented via a

matrix. One can combine matrices according to a set of

standard

MATRIX OPERATIONS

.

See also

EIGENVECTOR

;

GENERAL LINEAR GROUP

;

IDENTITY MATRIX

;

INVERSE MATRIX

.

matrix operations There are three basic arithmetic

operations one can perform on a

MATRIX

or on a pair

of matrices of the same dimension (that is, two matri-

ces with the same number of rows and the same num-

ber of columns).

1. Scalar Multiplication: To multiply a matrix by a real

number k, multiply each entry of the matrix by that

number. For example, if

then

In general, the formula for the (i,j)th element of the

scalar product kA is:

(kA)ij = kAij

2. Matrix Addition: To sum two matrices of the same

dimension, add corresponding entries and enter each

sum in the corresponding place in the matrix sum.

For example, if

and

then

In general, the formula for the (i,j)th element of the

sum A+ Bis:

(A+ B)ij = Aij + Bij