Page 343 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

334 Appendix I
AI ˆ IA (I:17)
Moreover, two diagonal matrices of the same order commute
0 10 1 0 1
0 0 0 0 0 0
a 1 b 1 a 1 b 1
B 0 a 2 0 CB 0 b 2 0 C B 0 a 2 b 2 0 C
AB ˆ B CB C ˆ B C
@ A@ A @ A
0 0 a n 0 0 b n 0 0 a n b n
ˆ BA (I:18)

Determinants
For a square matrix A, there exists a number called the determinant of the matrix. This
determinant is denoted by

a 11 a 12 a 1n

a 21 a 22
jAjˆ a 2n (I:19)

a n1 a n2 a nn
and is de®ned as the summation
X
jAjˆ ä P a 1i a 2 j a nq (I:20)
P
where ä P ˆÐ 1. The summation is taken over all possible permutations i, j, ... , q of
the sequence 1, 2, .. . , n. The value of ä P is ‡1(ÿ1) if the order i, j, .. . , q is
obtained by an even (odd) number of pair interchanges from the order 1, 2, .. . , n.
There are n! terms in the summation, half with ä P ˆ 1 and half with ä P ˆÿ1. Thus,
for a second-order determinant, we have

a 11
jAjˆ a 12 ˆ a 11 a 22 ÿ a 12 a 21 (I:21)

a 21 a 22
and for a third-order determinant, we have

a 11 a 12 a 13

jAjˆ a 21 a 22 a 23

a 31 a 32 a 33
ˆ a 11 a 22 a 33 ‡ a 12 a 23 a 31 ‡ a 13 a 21 a 32 ÿ (a 11 a 23 a 32 ‡ a 12 a 21 a 33 ‡ a 13 a 22 a 31 )(I:22)
If the determinant jAj of the matrix A vanishes, then the matrix A is said to be
singular. Otherwise, the matrix A is non-singular.
The determinant jAj has the following properties, which are easily derived from the
de®nition (I.20).
1. The interchange of any two rows or any two columns changes the sign of the
determinant.
2. Multiplication of all the elements in any row or in any column by a constant k gives
n
a new determinant of value kjAj. (Note that if B ˆ kA, then jBjˆ k jAj.)
3. The value of the determinant is zero if any two rows or any two columns are
identical, or if each element in any row or in any column is zero. As a special case
of properties 2 and 3, a determinant vanishes if any two rows or any two columns
are proportional.