Page 349 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

340 Appendix I

Trace
The trace Tr A of a square matrix A is de®ned as the sum of the diagonal elements
(I:59)
Tr A ˆ a 11 ‡ a 22 ‡ ‡ a nn
The operator Tr is a linear operator because
Tr(A ‡ B) ˆ (a 11 ‡ b 11 ) ‡ (a 22 ‡ b 22 ) ‡ ‡ (a nn ‡ b nn ) ˆ Tr A ‡ Tr B (I:60)
and
Tr(cA) ˆ ca 11 ‡ ca 22 ‡ ‡ ca nn ˆ c Tr A (I:61)
The trace of a product of two matrices, which may or may not commute, is
independent of the order of multiplication
n n n n
X X X X
Tr(AB) ˆ a ij b ji ˆ b ji a ij ˆ Tr(BA) (I:62)
iˆ1 jˆ1 jˆ1 iˆ1
Thus, the trace of the commutator [A, B] AB ÿ BA is equal to zero. Furthermore,
the trace of a continued product of matrices is invariant under a cyclic permutation of
the matrices
Tr(ABC Q) ˆ Tr(BC QA) ˆ Tr(C QAB) ˆ (I:63)
For a hermitian matrix, the trace is the sum of its eigenvalues
n
X
Tr A ˆ ë á (I:64)
áˆ1
To demonstrate the validity of equation (I.64), we ®rst take the trace of (I.58) to obtain
n
X
ÿ1
Tr(X AX) ˆ Tr Ë ˆ ë á (I:65)
áˆ1
We then note that
n n n n
X X X X
ÿ1 ÿ1 ÿ1
Tr(X AX) ˆ (X AX) áá ˆ X a ij X já
ái
áˆ1 áˆ1 iˆ1 jˆ1
n n n n n n
X X X ÿ1 X X X
ˆ a ij X já X ˆ a ij ä ij ˆ a ii ˆ Tr A (I:66)
ái
iˆ1 jˆ1 áˆ1 iˆ1 jˆ1 iˆ1
(á)
where X já ( x ) are the elements of X. Combining equations (I.65) and (I.66), we
j
obtain equation (I.64).