10. Minimal Complexity Realization of Structured Matrices
P. Dewilde · Society for Industrial and Applied Mathematics eBooks · 1999
10.1 INTRODUCTION The earlier chapters considered the class of matrices that satisfy displacement equations (cf. Ch. 1) and, hence, have small displacement ranks. There are also other kinds of structured matrices. As a general working definition, we propose “matrices whose entries satisfy class generic constraints that reduce the number of algebraically free parameters.” Class generic constraints on the entries can be of several kinds: • Linear constraints between entries. Examples are the following: (i) Toeplitz, Hankel, or even Cauchy matrices. (ii) Their generalizations to matrices of low displacement rank, as studied extensively by the Kailath school and its many ramifications (see Ch. 1). This is the class of matrices studied in the earlier chapters. • Hard value constraints on entries. Examples are the following: (i) Banded or multibanded matrices. (ii) Inverses of banded matrices and (possibly continuous) products of banded matrices with inverses of banded matrices. • Matrices described by a low-complexity time-varying state-space model. • Nonlinear algebraic constraints. (Unitary matrices may seem to be of this type, but they can be brought into the class with linear constraints via the transformation in which H is a Hermitian matrix and .)