Stability Matrices and the Solvability of Certain Systems of Linear Inequalities

Simon A. Levin

doi:10.1080/03081087408817068

Stability Matrices and the Solvability of Certain Systems of Linear Inequalities

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Abstract

Let A be a real n × n matrix with non-negative non-diagonal elements ai j (i ≠ J). A is called a stability matrix if and only if all of its eigenvalues have strictly negative real parts. It is proved that A is a stability matrix if and only if the system Ax ≽ 0, x ≽ 0 has no non-trivial solution. Further, one and only one of the systems Ax < 0, x > 0 and Ax ≽ 0, x ≽ 0 has a non-trivial solution.

Original language	English (US)
Pages (from-to)	253-255
Number of pages	3
Journal	Linear and Multilinear Algebra
Volume	2
Issue number	3
DOIs	https://doi.org/10.1080/03081087408817068
State	Published - Jan 1 1974
Externally published	Yes

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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10.1080/03081087408817068

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abstract = "Let A be a real n × n matrix with non-negative non-diagonal elements ai j (i ≠ J). A is called a stability matrix if and only if all of its eigenvalues have strictly negative real parts. It is proved that A is a stability matrix if and only if the system Ax ≽ 0, x ≽ 0 has no non-trivial solution. Further, one and only one of the systems Ax < 0, x > 0 and Ax ≽ 0, x ≽ 0 has a non-trivial solution.",

author = "Levin, {Simon A.}",

note = "Funding Information: t Research supported by NSF Grant GP33031X. $ Department of Theoretical and Applied Mechanics, Section of Ecology and Systematics, Center for Applied Mathematics.",

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N2 - Let A be a real n × n matrix with non-negative non-diagonal elements ai j (i ≠ J). A is called a stability matrix if and only if all of its eigenvalues have strictly negative real parts. It is proved that A is a stability matrix if and only if the system Ax ≽ 0, x ≽ 0 has no non-trivial solution. Further, one and only one of the systems Ax < 0, x > 0 and Ax ≽ 0, x ≽ 0 has a non-trivial solution.

AB - Let A be a real n × n matrix with non-negative non-diagonal elements ai j (i ≠ J). A is called a stability matrix if and only if all of its eigenvalues have strictly negative real parts. It is proved that A is a stability matrix if and only if the system Ax ≽ 0, x ≽ 0 has no non-trivial solution. Further, one and only one of the systems Ax < 0, x > 0 and Ax ≽ 0, x ≽ 0 has a non-trivial solution.

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Stability Matrices and the Solvability of Certain Systems of Linear Inequalities

Abstract

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