Page 348 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Matrices 339

The eigenvectors of a hermitian matrix with different eigenvalues are orthogonal. To
prove this statement, we consider two distinct eigenvalues ë 1 and ë 2 and their
(2)
corresponding eigenvectors x (1) and x , so that
Ax (1) ˆ ë 1 x (1) (I:52a)
Ax (2) ˆ ë 2 x (2) (I:52b)
If we multiply equation (I.52a) from the left by x (2) y and the adjoint of (I.52b) from the
(1)
right by x , we obtain
x (2) y Ax (1) ˆ ë 1 x (2) y x (1) (I:53a)

y (1)
(2) y (1)
(Ax ) x ˆ x (2) y A x ˆ x (2) y Ax (1) ˆ ë 2 x (2) y x (1) (I:53b)
where we have used equation (I.10) and noted that ë 2 is real. Subtracting equation
(I.53b) from (I.53a), we ®nd
(ë 1 ÿ ë 2 )x (2) y x (1) ˆ 0
Since ë 1 is not equal to ë 2 , we see that x (1) and x (2) are orthogonal.
The eigenvector x (á) corresponding to the eigenvalue ë á may be determined by
substituting the value for ë á into equation (I.47) and then solving the resulting
(á) (á) (á) (á)
simultaneous equations for the components x , x , ... , x n of x in terms of the
3
2
(á)
®rst component x . The value of the ®rst component is arbitrary, but it may be
1
speci®ed by requiring that the vector x (á) be normalized, i.e.,
(á) 2
(á) 2
(á) 2
x (á) y x (á) ˆjx j ‡jx j ‡ ‡ jx j ˆ 1 (I:54)
1 2 n
The determination of the eigenvectors for degenerate eigenvalues is somewhat more
complicated and is not discussed here.
We may construct an n 3 n matrix X using the n orthogonal eigenvectors x (á) as
columns
0 (1) (2) (n) 1
x x x
1 1 1
(1) (2)
x x x
B (n) C
X ˆ B 2 2 2 C (I:55)
@ A

x (1) x (2) x (n)
n n n
and a diagonal matrix Ë using the n eigenvalues
0 1
ë 1 0 0
B 0 ë 2 0 C
Ë ˆ B C (I:56)
@ A
0 0 ë n
Equation (I.47) may then be written in the form
AX ˆ XË (I:57)
The matrix X is easily seen to be unitary. Since the n eigenvectors are linearly
independent, the matrix X is non-singular and its inverse X ÿ1 exists. If we multiply
ÿ1
equation (I.57) from the left by X , we obtain
X AX ˆ Ë (I:58)
ÿ1
This transformation of the matrix A to a diagonal matrix is an example of a similarity
transform.