Page 341 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

332 Appendix I
0 1
a 11 b 11 ‡ a 12 b 21 ‡ a 13 b 31 a 11 b 12 ‡ a 12 b 22 ‡ a 13 b 32
AB ˆ @ a 21 b 11 ‡ a 22 b 21 ‡ a 23 b 31 a 21 b 12 ‡ a 22 b 22 ‡ a 23 b 32 A
a 31 b 11 ‡ a 32 b 21 ‡ a 33 b 31 a 31 b 12 ‡ a 32 b 22 ‡ a 33 b 32
Continued products, such as ABC, may be de®ned and evaluated if the matrices are
conformable. In such cases, multiplication is associative, for example
X X
ABC ˆ A(BC) ˆ (AB)C ˆ D; d il ˆ a ij b jk c kl (I:4)
j k
For the null matrix 0, all the matrix elements are zero
0 1
0 0 0
B 0 0 0 C
0 ˆ B C (I:5)
@ A

0 0 0
and we have
0 ‡ A ˆ A ‡ 0 ˆ A (I:6)
The product of an arbitrary m 3 n matrix A with a conformable n 3 p null matrix is
the m 3 p null matrix
A0 ˆ 0 (I:7)
In matrix algebra it is possible for the product of two conformable matrices, neither of
which is a null matrix, to be a null matrix. For example, if A and B are
0 1 0 1
1 1 0 1 ÿ1
A ˆ @ ÿ2 ÿ2 2 A ; B ˆ @ ÿ1 1 A
3 3 1 0 0
then the product AB is the 3 3 2 null matrix.
T
The transpose matrix A of a matrix A is obtained by interchanging the rows and
columns of A. If the matrix A is given by equation (I.1), then its transpose is
0 1
a 11 a 21 a m1
T B a 12 a 22 a m2 C
A ˆ B C (I:8)
@ A

a 1n a 2n a mn
T
T
T
Thus, the elements a of A are given by a ˆ a ji .
ij ij
Let the matrix C be the product of matrices A and B as in equation (I.3). The
T
elements c of the transpose of C are then given by
ik
n n n
X X X
T T T T T
c ˆ a kj b ji ˆ a b ˆ b a (I:9)
ik jk ij ij jk
jˆ1 jˆ1 jˆ1
where we have noted that a T ˆ a âá and b T ˆ b âá . Thus, we see that
áâ áâ
T
T
C ˆ B A T
or
T
T
(AB) ˆ B A T (I:10)
This result may be generalized to give
T
T
T
(AB Q) ˆ Q B A T (I:11)
as long as the matrices are conformable.

If each element a ij in a matrix A is replaced by its complex conjugate a , then the
ij

resulting matrix A is called the conjugate of A. The transposed conjugate of A is