Trace norm bounds for stable Lyapunov operators
type: Item
instance of: scholarly article
title: Trace norm bounds for stable Lyapunov operators
zbMATH Open document ID: 0826.15009
DOI: 10.1016/0024-3795(93)00219-P
author: Gary Hewer
author: Charles S. Kenney
published in: Linear Algebra and its Applications
publication date: 1995-12-03T00:00:00Z
review text: In the introduction the authors note that Lyapunov equations of the form \(A^TX + XA = - Q\) have been extensively studied because of their deep connections with dynamical systems, control theory, and a variety of other areas. The lemma in Section 1 illustrates one of the basic methods of this paper: first apply the trace to an integral representation of the Lyapunov solution, and then interchange the trace and integral operators. Next, reverse the order of the matrices inside the trace, and finally take the trace outside the integral. In discussing dual norms in Section 2, they follow the presentation of \textit{R. Horn} and \textit{C. R. Johnson} [Topics in matrix analysis (1991; Zbl 0729.15001)]. Dual norms arise naturally in the area of quantum mechanics in showing that any unitarily invariant matrix norm is associated with a symmetric gauge function and vice versa. They end this section with a look at the problem of bounding the Frobenius norm of the inverse Lyapunov operator \(\Phi\). The trace results of the previous sections find useful application in the problem of estimating the sensitivity of solutions to the Riccati equation \(0 = G + A^TX + XA - XFX\) in Section 3. In the previous section they show that the power method gives upper and lower bounds on the Frobenius norm of \(\Phi\). The same approach gives upper and lower bounds on the Frobenius norm of the operator \(\Pi (n) : = \Phi (XMX)\).
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 769044
zbMATH Keywords: trace norm bounds
zbMATH Keywords: Lyapunov equations
zbMATH Keywords: dynamical systems
zbMATH Keywords: trace operator
zbMATH Keywords: dual norms
zbMATH Keywords: symmetric gauge function
zbMATH Keywords: inverse Lyapunov operator
zbMATH Keywords: Riccati equation
zbMATH Keywords: power method
zbMATH Keywords: Frobenius norm
reviewed by: Yueh-er Kuo
describes a project that uses: Algorithm 432
MaRDI profile type: Publication
cites work: Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4]
cites work: A LINPACK-style condition estimator for the equation&lt;tex&gt;AX-XB^{T} = C&lt;/tex&gt;
cites work: Q3702392
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cites work: The Sensitivity of the Stable Lyapunov Equation
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cites work: Explicit Solutions of Linear Matrix Equations
cites work: On norm bounds for algebraic Riccati and Lyapunov equations
cites work: Q5751863
cites work: Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems
cites work: On the Separation of Two Matrices
cites work: Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equation
cites work: Generalizations of theorems of Lyapunov and Stein
cites work: Solving the Matrix Equation $\sum _{\rho = 1}^r f_\rho (A)Xg_\rho (B) = C$
Wikidata QID: Q126857649
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description: scientific article; zbMATH DE number 769044