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Stability and inertia. (English) Zbl 0972.15009
This is an excellent, clear and concise presentation of matrix stability and inertia theory, including generalized stability, and the application of these concepts to solutions of continuous and discrete problems. Succinct proofs and several open problems are given. The extensive bibliography documents the history and development of this subject, bringing together references from economics, control theory, mechanics, numerical analysis, and matrix theory.
The Bézoutian and the dual concepts of controllability and observability are discussed and exploited. The classic Lyapunov and Stein theorems are proved, which characterize the stability of \(x'(t)= Ax(t)\) and \(x(k+1)= Ax(k)\) in terms of solutions to matrix equations. Subsequent extensions and related results are presented, bringing the theory up to date, and the interrelationships between theorems are explored.
Some applications of inertia theory are presented, including root separation, D-stability, structural analysis and continued fractions. Numerical methods for determining stability and inertia are also discussed.

MSC:
15A42 Inequalities involving eigenvalues and eigenvectors
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
15A24 Matrix equations and identities
93C05 Linear systems in control theory
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