Page 345 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

336 Appendix I

It follows from equation (I.27) that the product of two non-singular matrices is also
non-singular.

Special square matrices
^
The adjugate matrix A of the square matrix A is de®ned as
0 1
C 11 C 21 C n1
^ B C 12 C 22 C n2 C
A ˆ B C (I:28)
@ A
C 1n C 2n C nn
where C ij are the cofactors of the elements a ij of the determinant jAj of A. Note that
^
^
the element ^ a kl of A is the cofactor C lk . The matrix product AA is a matrix B whose
elements b ij are given by
n n
X X
b ij ˆ a ik ^ a kj ˆ a ik C jk ˆjAjä ij (I:29)
kˆ1 kˆ1
where equation (I.26) was introduced. Thus, we have
^
^
AA ˆ B ˆjAjI ˆ AA (I:30)
^
where I is the unit matrix in equation (I.16), and we see that the matrices A and A
commute.
Any non-singular square matrix A possesses an inverse matrix A ÿ1 de®ned as
^
A ÿ1 ˆ A=jAj (I:31)
From equation (I.30) we observe that
ÿ1
AA ÿ1 ˆ A A ˆ I (I:32)
Consider three square matrices A, B, C such that AB ˆ C. Then we have
ÿ1
ÿ1
A AB ˆ A C
or
ÿ1
B ˆ A C (I:33)
Thus, the inverse matrix plays the role of division in matrix algebra. Multiplication of
equation (I.33) from the left by B ÿ1 and from the right by C ÿ1 yields
ÿ1
C ÿ1 ˆ B A ÿ1
or
ÿ1
(AB) ÿ1 ˆ B A ÿ1 (I:34)
This result may easily be generalized to show that
ÿ1
(AB Q) ÿ1 ˆ Q ÿ1 B A ÿ1 (I:35)
A square matrix A is hermitian or self-adjoint if it is equal to its adjoint, i.e., if

A ˆ A or a ij ˆ a . Thus, the diagonal elements of a hermitian matrix are real.
y
ji
A square matrix A is orthogonal if it satis®es the relation
T
T
AA ˆ A A ˆ I
ÿ1
T
If we multiply AA ˆ I from the left by A , then we have the equivalent de®nition
T
A ˆ A ÿ1
T
Since the determinants jAj and jA j are equal, we have from equation (I.27)
2
jAj ˆ 1 or jAjˆÐ 1