Controllability, inertia, and stability for tridiagonal matrices
type: Item
instance of: scholarly article
title: Controllability, inertia, and stability for tridiagonal matrices
zbMATH Open document ID: 0531.15008
DOI: 10.1016/0024-3795(84)90126-5
published in: Linear Algebra and its Applications
publication date: 1984-01-01T00:00:00Z
review text: If A is an \(n\times n\) complex tridiagonal matrix and \(In A=(\pi(A),\nu(A),\delta(A))\) is the inertia of A (\(\pi\),\(\nu\),\(\delta\) being the number of eigenvalues with positive, negative and zero real parts, respectively), A is called positive stable (or just stable) if \(\pi(A)=n\) and D-stable if DA is stable for every positive diagonal matrix D. On the other hand, if B is an \(n\times n\) complex matrix, the pair (A,B) is controllable if \([B,AB,A^ 2B,...,A^{n-1}B]\) has rank n. In a previous paper the author, \textit{B. Datta} and \textit{C. Johnson} [A semi-definite Lyapunov theorem and the characterization of tridiagonal D- stable matrices. SIAM J. Algebra Discrete Math., (to appear)] characterized the real tridiagonal matrices which are D-stable. Extending the same techniques and using also results of other mathematicians about inertia and stability in relationship with controllability, the author gives criteria for the controllability of certain pairs of tridiagonal matrices, obtains some inertia results, characterizes certain matrices which are positive stable or D-stable.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 3842060
zbMATH Keywords: Chen-Wimmer theorem
zbMATH Keywords: complex tridiagonal matrix
zbMATH Keywords: inertia
zbMATH Keywords: positive stable
zbMATH Keywords: D-stable
zbMATH Keywords: controllability
author: David H. Carlson
MaRDI profile type: Publication
full work available at URL: 0024-3795(84)90126-5
OpenAlex ID: W1977540954
cites work: Some Applications of the Lyapunov Matrix Equation
cites work: A Semi-Definite Lyapunov Theorem and the Characterization of Tridiagonal D-Stable Matrices
cites work: The Lyapunov matric equation SA+A*S=S*B*BS
cites work: A Generalization of the Inertia Theorem
cites work: A Stability Theorem for a Class of Damped Dynamic Systems and some Applications
cites work: Q5582663
cites work: Ein Verfahren zur Stabilitätsfrage bei Matrizen-Eigenwertproblemen
cites work: On real eigenvalues of real tridiagonal matrices
cites work: Inertia theorems for matrices, controllability, and linear vibrations
cites work: An inertia theorem for tridiagonal matrices and a criterion of Wall on continued fractions
description: scientific article; zbMATH DE number 3842060