Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.4 BASIC MATRIX ALGEBRA

Now, sincexis arbitrary, we can immediately deduce the way in which matrices

are added or multiplied, i.e.

(A+B)ij=Aij+Bij, (8.26)

(λA)ij=λAij, (8.27)

(AB)ij=

∑

k

AikBkj. (8.28)

We note that a matrix element may, in general, be complex. We now discuss

matrix addition and multiplication in more detail.

8.4.1 Matrix addition and multiplication by a scalar

From (8.26) we see that the sum of two matrices,S=A+B, is the matrix whose

elements are given by

Sij=Aij+Bij

for every pair of subscriptsi, j, withi=1, 2 ,...,Mandj=1, 2 ,...,N.For

example, ifAandBare 2×3 matrices thenS=A+Bis given by

(

S 11 S 12 S 13

S 21 S 22 S 23

)

=

(

A 11 A 12 A 13

A 21 A 22 A 23

)

+

(

B 11 B 12 B 13

B 21 B 22 B 23

)

=

(

A 11 +B 11 A 12 +B 12 A 13 +B 13

A 21 +B 21 A 22 +B 22 A 23 +B 23

)

. (8.29)

Clearly, for the sum of two matrices to have any meaning, the matrices must have

the same dimensions, i.e. both beM×Nmatrices.

From definition (8.29) it follows thatA+B=B+Aand that the sum of a

number of matrices can be written unambiguously without bracketting, i.e. matrix

addition iscommutativeandassociative.

The difference of two matrices is defined by direct analogy with addition. The

matrixD=A−Bhas elements

Dij=Aij−Bij, fori=1, 2 ,...,M,j=1, 2 ,...,N. (8.30)

From (8.27) the product of a matrixAwith a scalarλis the matrix with

elementsλAij, for example

λ

(

A 11 A 12 A 13

A 21 A 22 A 23

)

=

(

λA 11 λA 12 λA 13

λA 21 λA 22 λA 23

)

. (8.31)

Multiplication by a scalar is distributive and associative.