Adaptive Filters

CHAPTER B Linear Algebra

Throughout the book, the reader will be exposed to a variety of concepts from linear algebra and matrix theory in a motivated manner. In this way, after progressing sufficiently enough into the book, readers will be able to master many of the useful concepts described herein.

B.1 HERMITIAN AND POSITIVE-DEFINITE MATRICES

Hermitian matrices. The Hermitian conjugate, A*, of a matrix A is the complex conjugate of its transpose, e.g.,

A Hermitian matrix is a square matrix satisfying A* = A, eg.,

so that A is Hermitian.

Spectral decomposition. Hermitian matrices can only have real eigenvalues. To see this, assume u_i is an eigenvector of A corresponding to an eigenvalue λ_i, i.e., Au_i = λ_iu_i. Multiplying from the left by we get , where || • || denotes the Euclidean norm of its argument. Now the scalar quantity on the left-hand side of this equality is real since it coincides with its complex conjugate, namely . Therefore, λ_i must be real too.

Another important property ...