On a new class of realization formulas and their application
type: Item
instance of: scholarly article
title: On a new class of realization formulas and their application
zbMATH Open document ID: 0858.15012
DOI: 10.1016/0024-3795(95)00602-8
author: Daniel Alpay
author: Harry Dym
published in: Linear Algebra and its Applications
publication date: 1997-03-23T00:00:00Z
review text: This paper is devoted to derive and develop the theory of a new set of realization formulas for a class of matrix-valued meromorphic functions \(W(\lambda)\), which includes the class of rational matrix-valued functions. The realization formulas for a general class of \(p\times q\) matrix-valued functions meromorphic in a connected nonempty open set \(\Omega\) of the complex plane are first established together with a number of auxiliary formulas and results. Then the authors discuss the concept of observability, controllability and minimality, and give a number of equivalent characterizations of each. The similarity of two minimal realizations is also discussed. Next, they develop the necessary and sufficient conditions in terms of Lyapunov-Stein-type equations for the existence of the realizations of this type, which are either isometric or coisometric (or both) matrix-valued functions with respect to a pair (or just one) signature matrices on the boundary of an appropriately defined region. Subsequently, they discuss pairings wherein a given \(W(\lambda)\) is ``paired'' with a partner \(W_\pi(\lambda)\), which is often an effective substitute for \(W(\lambda)^*\) when the conditions for \(W(\lambda)\) to be either isometric or coisometric (with respect to two given signature matrices) on the boundary fail. Finally, they devote to embedding the basic Lyapunov-Stein-type equations into a larger array of equations and related implications, and besides, factorization formulas for the pair space considered and minimal \(J\) unitary factorizations of \(W (\lambda)\) are treated.
reviewed by: Yang Yingchen
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 93B15
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93B07
zbMATH DE Number: 931881
zbMATH Keywords: reproducing kernel space
zbMATH Keywords: \(J\) unitary factorization
zbMATH Keywords: realization formulas
zbMATH Keywords: matrix-valued meromorphic functions
zbMATH Keywords: rational matrix-valued functions
zbMATH Keywords: observability
zbMATH Keywords: controllability
zbMATH Keywords: minimality
zbMATH Keywords: Lyapunov-Stein-type equation
zbMATH Keywords: signature matrices
MaRDI profile type: Publication
full work available at URL: 0024-3795(95)00602-8
OpenAlex ID: W2059828775
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description: scientific article; zbMATH DE number 931881