Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

Appendix D 275
I Hermitian conjugate (“transpose conjugate”) of a matrix: A †
To obtain the Hermitian conjugate (or “transpose conjugate” or “adjoint”) A † of the
matrix A, take the complex conjugate and transpose:
† = ¡ ∗¢ = ¡ ¢ ∗
(D.11)
where ¡
† ¢ =( ) ∗ (D.12)
I Hermitian matrices: AmatrixA is Hermitian, if
A † =(A ∗ ) = A
(D.13)
i.e., ¡
† ¢ =( ) ∗ = (D.14)
I Inverse of a matrix: A −1
AmatrixA −1 is the inverse of the original matrix A, if
AA −1 = A −1 A = I
(D.15)
I Orthogonal matrices: AmatrixA is orthogonal, iftheinverseofA equals its transpose:
A −1 = A
(D.16)
I Unitary matrices: AmatrixU is unitary, iftheinverseofU equals its transpose
conjugate:
U −1 = U † =(U ∗ )
(D.17)
y
U † U = I
(D.18)
I
Trace of a matrix:
tr (A) =
X
=1
(D.19)
I
Matrix addition:
C = A + B
(D.20)
with
= +
(D.21)