Page 346 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Matrices 337

The product of two orthogonal matrices is an orthogonal matrix as shown by the
following sequence
T
T
T
ÿ1
(AB) ˆ B A ˆ B A ÿ1 ˆ (AB) ÿ1
where equations (I.10) and (I.34) were used. The inverse of an orthogonal matrix is
also an orthogonal matrix as shown by taking the transpose of A ÿ1 and noting that the
order of transposition and inversion may be reversed
ÿ1 ÿ1
T ÿ1
ÿ1 T
(A ) ˆ (A ) ˆ (A )
y
A square matrix A is unitary if its inverse is equal to its adjoint, i.e., if A ÿ1 ˆ A or

y
y
if AA ˆ A A ˆ I. For a real matrix, with all elements real so that a ij ˆ a , there is
ij
no distinction between an orthogonal and a unitary matrix. In that case, we have
T
ÿ1
y
A ˆ A ˆ A .
Linear vector space
Avector x in three-dimensional cartesian space may be represented as a column
matrix
0 1
x 1
x ˆ @ x 2 A (I:36)
x 3
where x 1 , x 2 , x 3 are the components of x. The adjoint of the column matrix x is a row
matrix

y
x ˆ (x x x ) (I:37)
1
2
3
The scalar product of the vectors x and y when expressed in matrix notation is
0 1
y 1

y
x y ˆ (x x x ) y 2 A ˆ x y 1 ‡ x y 2 ‡ x y 2 (I:38)
@
1 2 3 1 2 2
y 3
Consequently, the magnitude of the vector x is
2 1=2
2
2
(x x) 1=2 ˆ (jx 1 j ‡jx 2 j ‡jx 3 j ) (I:39)
y
If the vectors x and y are orthogonal, then we have x y ˆ 0. The unit vectors i, j, k in
y
matrix notation are
0 1 0 1 0 1
1 0 0
0
0
1
i ˆ @ A ; j ˆ @ A ; k ˆ @ A (I:40)
0 0 1
^
A linear operator A in three-dimensional cartesian space may be represented as a
^
3 3 3 matrix A with elements a ij . The expression y ˆ Ax in matrix notation becomes
0 1 0 10 1 0 1
y 1 a 11 a 12 a 13 x 1 a 11 x 1 ‡ a 12 x 2 ‡ a 13 x 3
y ˆ @ y 2 A ˆ Ax ˆ @ a 21 a 22 a 23 A@ x 2 A ˆ @ a 21 x 1 ‡ a 22 x 2 ‡ a 23 x 3 A
y 3 a 31 a 32 a 33 x 3 a 31 x 1 ‡ a 32 x 2 ‡ a 33 x 3
(I:41)
If A is non-singular, then in matrix notation the vector x is related to the vector y by
ÿ1
x ˆ A y (I:42)
The vector concept may be extended to n-dimensional cartesian space, where we
have n mutually orthogonal axes. Each vector x then has n components (x 1 , x 2 , .. . ,