18.3 Matrix algebra 507
Matrix multiplication
The matrix productC 1 = 1 AB(in this order, with Ato the left of B) is defined only if
the number of columns of A 1 = 1 the number of rows of B (18.25)
Then, if Ais anm 1 × 1 nmatrix with elementsa
ijandBis ann 1 × 1 pmatrix with elements
b
ij, the productC 1 = 1 ABis anm 1 × 1 pmatrix whose elements are
(18.26)
In the simplest case, if ais a row vector (matrix) with ncomponents a
iand bis a
column vector (matrix) with ncomponentsb
i, the product abis
(18.27)
The product is a number, a 11 × 11 matrix. In this case the matrix product corresponds
to the scalar producta 1
·
1 bof two (n-dimensional) vectors (see sections 16.5 and 16.10).
In the general case, the prescription (18.26) for the ijth element is the ‘scalar product’
of the ith row of Aand the jth column of B. Thus, with the relevant row and column
in boldface,
(18.28)
=
cc c c
cc c c
cc
jpjpi
i11
121121 22221
2...............
c
ij............
c
cc c c
ipm 1 m 2 mj mpCAB
1
2==
aaa a
aaa a
nn11
12 13121 22 232......
aaa
i
ii 33......
a
inaaa a
m
mm1 mn
23bb b
bb b
11 p
12121 22............b
b
jj12 2231 3221 2ppn n npbb b
bb b
............
b
b
jnj3ab=
()aaa a
b
b
b
b
nn123123==+ + ++ =
∑
ab ab ab ab ab
nnkn11 2 2 33 kk1cabab ab ab ab
ij i j i j i j in njknik=+ + ++ =
=∑
11 2 2 331kkj