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On a quadratic matrix equation associated with an \(M\)-matrix. (English) Zbl 1017.15004

The main purpose of this work is to study the quadratic matrix equation \( X^{2}-EX-F=0\) where \(E,F,X\in \mathbb{R}^{n\times n}\), \(E\) is diagonal and \(F\) is an \( M\)-matrix. The main approach that the authors are using for solving the above quadratic matrix equation, is by transforming it into an equation that belongs to a special class of non-symmetric algebraic Riccati equations (AREs). Existence and uniqueness of \(M\)-matrix solutions, as well as numerical methods for finding the desired \(M\)-matrix solution are given. The theory has been illustrated via an application to noisy Wiener-Hopf problems. The paper ends with some discussions concerning the general non-symmetric ARE.

MSC:

15A24 Matrix equations and identities
15B48 Positive matrices and their generalizations; cones of matrices
65F30 Other matrix algorithms (MSC2010)
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
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