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Article: 6 Appendix.(Majorization and Matrix-Monotone Functions in Wireless Communications)
- Article from:
- Foundations and Trends in Communications and Information Theory
- Article date:
- December 15, 2006
- Author:
CopyrightCOPYRIGHT 2006 Now Publishers, Inc. This material is published under license from the publisher through the Gale Group, Farmington Hills, Michigan. All inquiries regarding rights should be directed to the Gale Group. (Hide copyright information)
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6.1 Linear Algebra
Most of the material can be found in [50, 51]. Since this section contributes basic results, the proofs are omitted.
Proposition 6.1 (Singular value decomposition). Every square matrix A of dimension n can be decomposed into
A = V[summation][W.sup.H],
where V, W are unitary square matrices of size n x n and S is a diagonal matrix with nonnegative main diagonal entries and the rank of S is the same as the rank of A.
Proposition 6.2 (Eigenvalue decomposition). Let A be a square matrix of dimension n. Then A is Hermitian if and only if there is a square unitary matrix U of size n and a real diagonal matrix ...