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Singular-value-like decomposition for complex matrix triples

Please always quote using this URN:urn:nbn:de:0296-matheon-4224
  • The classical singular value decomposition for a matrix $A\in\Cmn$ is a canonical form for $A$ that also displays the eigenvalues of the Hermitian matrices $AA^\ast$ and $A^\ast A$. In this paper, we develop a corresponding decomposition for $A$ that provides the Jordan canonical forms for the complex symmetric matrices $AA^T$ and $A^TA$. More generally, we consider the matrix triple $(A,G_1,G_2)$, where $G_1\in\CC{m}, G_2\in\CC{n}$ are invertible and either complex symmetric and complex skew-symmetric, and we provide a canonical form under transformations of the form $(A,G_1,G_2)\mapsto(X^T A Y, X^T G_1X, Y^T G_2Y)$, where $X,Y$ are nonsingular.

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Metadaten
Author:Christian Mehl, Volker Mehrmann, Hongguo Xu
URN:urn:nbn:de:0296-matheon-4224
Referee:Volker Mehrmann
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2007/10/24
Release Date:2007/12/10
Tag:
Institute:Technische Universität Berlin
MSC-Classification:15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A21 Canonical forms, reductions, classification
34-XX ORDINARY DIFFERENTIAL EQUATIONS / 34Axx General theory / 34A30 Linear equations and systems, general
65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra / 65F15 Eigenvalues, eigenvectors
65-XX NUMERICAL ANALYSIS / 65Lxx Ordinary differential equations / 65L05 Initial value problems
65-XX NUMERICAL ANALYSIS / 65Lxx Ordinary differential equations / 65L80 Methods for differential-algebraic equations
93-XX SYSTEMS THEORY; CONTROL (For optimal control, see 49-XX) / 93Bxx Controllability, observability, and system structure / 93B40 Computational methods
Preprint Number:412
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