Positive diagonal solutions to the Lyapunov equations
- 1 January 1978
- journal article
- research article
- Published by Taylor & Francis in Linear and Multilinear Algebra
- Vol. 5 (4) , 249-256
- https://doi.org/10.1080/03081087808817203
Abstract
We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DAt positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions.Keywords
This publication has 9 references indexed in Scilit:
- Application of M-matrices to the stability problems of composite dynamical systemsJournal of Mathematical Analysis and Applications, 1975
- Sufficient conditions for D-stabilityJournal of Economic Theory, 1974
- Cones, Matrices and Mathematical ProgrammingPublished by Springer Nature ,1973
- More on linear inequalities with applications to matrix theoryJournal of Mathematical Analysis and Applications, 1971
- Inertia theory for simultaneously triangulable complex matricesLinear Algebra and its Applications, 1969
- Positive operators and an inertia theoremNumerische Mathematik, 1965
- A Generalization of a Theorem of LyapunovJournal of the Society for Industrial and Applied Mathematics, 1961
- Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer SystemePublished by Walter de Gruyter GmbH ,1960
- Some general theorems on iterantsJournal of Research of the National Bureau of Standards, 1952