On Nonnegative Solutions of the Equation $AD + DA' = - C$
01 May 1970-Siam Journal on Applied Mathematics (Society for Industrial and Applied Mathematics)-Vol. 18, Iss: 3, pp 704-714
About: This article is published in Siam Journal on Applied Mathematics.The article was published on 1970-05-01. It has received 87 citations till now.
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14 Mar 2005
TL;DR: This book brings together a vast body of results on matrix theory for easy reference and immediate application with hundreds of identities, inequalities, and matrix facts stated rigorously and clearly.
Abstract: "Matrix Mathematics" is a reference work for users of matrices in all branches of engineering, science, and applied mathematics. This book brings together a vast body of results on matrix theory for easy reference and immediate application. Each chapter begins with the development of relevant background theory followed by a large collection of specialized results. Hundreds of identities, inequalities, and matrix facts are stated rigorously and clearly with cross references, citations to the literature, and illuminating remarks. Twelve chapters cover all of the major topics in matrix theory: preliminaries; basic matrix properties; matrix classes and transformations; matrix polynomials and rational transfer functions; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; norms; functions of matrices and their derivatives; the matrix exponential and stability theory; and linear systems and control theory. A detailed list of symbols, a summary of notation and conventions, an extensive bibliography with author index, and an extensive index are provided for ease of use. The book will be useful for students at both the undergraduate and graduate levels, as well as for researchers and practitioners in all branches of engineering, science, and applied mathematics.
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...Solutions of the Lyapunov equation under weak conditions are considered in [512]....
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TL;DR: The continuous-time Lyapunov equation has a unique Hermitian solution, X, if and only if Xt + X~j ^ 0 for all i and j (Barnett, 1975) as discussed by the authors.
Abstract: is called the continuous-time Lyapunov equation and is of interest in a number of areas of control theory such as optimal control and stability (Barnett, 1975; Barnett & Storey, 1968). The equation has a unique Hermitian solution, X, if and only if Xt + X~j ^ 0 for all i and j (Barnett, 1975). In particular if every Xt has a negative real part, so that A is stable, and if C is non-negative definite then X is also non-negative definite (Snyders & Zakai, 1970; Givens, 1961; Gantmacher, 1959). In this case, since X is nonnegative definite, it can be factorized as
406 citations
TL;DR: In this paper, a closed-form finite series representation of the unique solution X of the matrix equation AX − XB=C is developed, where the image, kernel, and rank of X are related to the controllable and unobservable subspaces of the (A, C) and (C, B) pairs respectively.
189 citations
TL;DR: In this paper, the eigenvalues of the solutions to a class of continuous and discrete-time Lyapunov equations with symmetric coefficient matrices and right-hand side matrices of low rank were studied.
185 citations
TL;DR: Two efficient methods for solving generalized Lyapunov equations and their implementations in FORTRAN 77 are presented and the first one is a generalization of the Bartels–Stewart method and the second is an extension of Hammarling's method to generalized Lyapsi equations.
Abstract: Two efficient methods for solving generalized Lyapunov equations and their implementations in FORTRAN 77 are presented. The first one is a generalization of the Bartels–Stewart method and the second is an extension of Hammarling's method to generalized Lyapunov equations. Our LAPACK based subroutines are implemented in a quite flexible way. They can handle the transposed equations and provide scaling to avoid overflow in the solution. Moreover, the Bartels–Stewart subroutine offers the optional estimation of the separation and the reciprocal condition number. A brief description of both algorithms is given. The performance of the software is demonstrated by numerical experiments.
171 citations
Cites background from "On Nonnegative Solutions of the Equ..."
...Under this assumption equations (1) and (2) are equivalent to the Lyapunov equations (AE 1)TX +X(AE 1) = E TY E 1 (5) and (AE 1)TX(AE 1) X = E TY E 1; (6) respectively, and the classical result about the positive de nite solution of the stable Lyapunov equation [14] remains valid for the generalized equations....
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