文档介绍：Supelec
Random Matrix Theory
for
munications
Merouane´ Debbah
merouane.******@
February, 2008
Presentation
Definitions, Probability and convergence measures
1
Circularly Symmetric
We say that a random vector x plex entries is circularly symmetric when
E[xxT ] = 0 even when E[xxH] 6= 0.
2
Eigenvalues
P. Lancaster, ”Theory of Matrices”, Academic Press, NY, 1969.
plex matrix H can be written:
H = URUH
where U is unitary (U−1 = UH) and R is an upper triangular matrix.
The eigenvalues of H are the principal diagonal elements of R. If H is hermitian
(H = H∗), then R is diagonal and the eigenvalues are real.
3
Rayleigh Quotient
Definition. The extreme eigenvalues of a Hermitian matrix H can be characterized in
terms of the Rayleigh quotient R(x) defined by:
xHHx
R(x) =
xHx
Lemma. Given a Hermitian matrix H, let λmin and λmax denote respectively the
minimum and maximum eigenvalues of H. Then
λmin = minxR(x)
λmax = maxxR(x)
4
Norm of a matrix
Definition. A norm T (H) on the space of N × N matrices satisfies the following
properties:
(1) T (H) ≥ 0 with equality if and only if H = 0 is the all zero matrix.
(2) For any two matrices H1 and H2,
T (H1 + H2) ≤ T (H1) + T (H2)
(3) For any scalar α and matrix H,
T (αH) =| α| T (H)
5
Hilbert-Schmidt norm
For an N × plex matrix W = (wij), the Hilbert-Schmidt norm (or Schur norm or
Euclidean norm or Frobenius norm) of W is defined as:
sX
2
|| W ||= | wij |
ij
6
Useful Inequalities
1. | Trace(BC) |≤|| B |||| C ||
2. If U is a unitary matrix, then for any C of the same order,
|| CU ||=|| UC ||=|| C ||
3. For a Hermitian matrix B with eigenvalues λ1, ..., λN , and any C,
max(|| BC ||, || CB ||) ≤ max | λi ||| C ||
1≤i≤N
7
Useful Inequalities
1. For a rectangular matrix A and B of the same size,
rank(A + B) ≤ rank(A) + rank(B)
2. For rectangular matrices A and B in which AB is defined:
rank(AB) ≤ min(rank(A), rank(B))
3. For Hermitian N × N mat