B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

Appendix D: Basic Matrix Properties and Operations 883
Based on the definition of determinant, for a scalar a,
det (aA) = a n det (A)
det (AT) = det (A)
(D.lla)
(D.llb)
For an identity matrix
det (I) = 1
(D. l lc)
Also, for two square matrices A and B,
det (AB) = det (A ) det (B)
(D.lld)
Therefore,
(D.lle)
For an m x n matrix A and an n x m matrix B, we have
det (l m +AB) = det (l n + BA)
(D. 12)
D.5 Trace
The trace of square matrix A is the sum of its diagonal entries
Tr(A) = I>i,i
i=l
(D.13)
For an m x n matrix A and an n x m matrix B, we have
Tr (AB) = Tr (BA)
D.6 Eigendecomposition
If the n x n square matrix A is Hermitian, then the equation
Au = AU
(D. 14)
(D. 15)
specifies an eigenvalue A and the associated eigenvector u.
When A is Hermitian, its eigenvalues are real-valued. Furthermore, A can be decomposed
into
A= UAU H (D. 16)
in which the matrix
U = [u1 u2 · · · Un ] (D. 17)
consists of orthogonal eigenvectors such that
(D.18)
Matrices satisfying this property are called unitary.