Page 342 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Matrices 333
y
1
called the adjoint of A and is denoted by A . The elements a of A are obviously
y
y
ij
y

given by a ˆ a .
ji
ij
Square matrices
Square matrices are of particular interest because they apply to many physical
situations.
A square matrix of order n is symmetric if a ij ˆ a ji ,(i, j ˆ 1, 2, ... , n), so that
T
T
A ˆ A , and is antisymmetric if a ij ˆÿa ji ,(i, j ˆ 1, 2, .. . , n), so that A ˆÿA .
The diagonal elements of an antisymmetric matrix must all be zero. Any arbitrary
square matrix A may be written as the sum of a symmetric matrix A (s) and an
antisymmetric matrix A (a)
A ˆ A (s) ‡ A (a) (I:12)
where
(s) 1 (a) 1
a ˆ (a ij ‡ a ji ); a ˆ (a ij ÿ a ji ) (I:13)
ij 2 ij 2
A square matrix A is diagonal if a ij ˆ 0for i 6ˆ j. Thus, a diagonal matrix has the
form
0 1
0 0
a 1
B 0 a 2 0 C
A ˆ B C (I:14)

@ A
0 0 a n
A diagonal matrix is scalar if all the diagonal elements are equal, a 1 ˆ a 2 ˆ ˆ
a n a, so that
0 1
a 0 0
B 0 a 0 C
A ˆ B C (I:15)

@ A
0 0 a
A special case of a scalar matrix is the unit matrix I, for which a equals unity
0 1
1 0 0
B 0 1 0 C
I ˆ B C (I:16)

@ A
0 0 1
The elements of the unit matrix are ä ij , the Kronecker delta function.
For square matrices in general, the product AB is not equal to the product BA.For
example, if

0 1 2 0
A ˆ ; B ˆ
1 0 0 3
then we have

0 3 0 2
AB ˆ ; BA ˆ 6ˆ AB
2 0 3 0
If the product AB equals the product BA, then A and B commute. Any square matrix
A commutes with the unit matrix of the same order
1 Mathematics texts use the term transpose conjugate for this matrix and apply the term adjoint to the
adjugate matrix de®ned in equation (I.28).