Equivalence Classes of Hermitian Matrices and Their Schur Parametrization
Philippe Delsarte, Yves V. Genin, Yves Kamp · SIAM Journal on Algebraic and Discrete Methods · 1983
A natural equivalence relation on Hermitian matrices is introduced to analyze the concepts of Toeplitz distance and displacement rank. Two Hermitian block-matrices are said to be equivalent when they are congruent under a block-triangular Toeplitz transformation. The matrices having the smallest Toeplitz distance within a given equivalence class are identified. This minimum distance equals the displacement rank minus twice the block size. Some $\Sigma $-unitary transfer functions $S( z )$ and some Schur-like functions $\Phi ( z )$ are constructed from the Lyapunov relation defining the displacement rank. These functions are used to characterize the equivalence classes and especially their minimum-distance representatives. A canonical factorization of $S( z )$ corresponds to a Schur-like decomposition of $\Phi ( z )$, involving a sequence of generalized Schur parameters $E_k $. The sets of functions $S( z )$, $\Phi ( z )$ and of Schur sequences $( E_k )$ characterizing an equivalence class are described in detail. Some results are obtained concerning the Schur parameters of the inverse of a given matrix.