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Factoring spectral matrices in linear-quadratic models. (English) Zbl 0725.15015

Let \(\Gamma =\Gamma (z)\) be a matrix function where \(z\) is on the unit circle, and assume \(\Gamma^*=\Gamma\). The authors give an algorithm to find a matrix function \(C=C(z)\) that is analytic and invertible on the unit disk and such that \(CC^*=\Gamma\), provided there is a not necessarily invertible matrix function \(F=F(z)\) such that \(FF^*=\Gamma\). This problem arises in linear-quadratic stochastic control problems.

MSC:

15A23 Factorization of matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
15A54 Matrices over function rings in one or more variables
49N10 Linear-quadratic optimal control problems
93E20 Optimal stochastic control
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References:

[1] Ball, Joseph A., Israel Gohberg and Leiba Rodman, Realization and interpolation of rational matrix functions, in: Operator theory: Advances and applications, Vol. 33 (Birkhauser Verlag, Basel).; Ball, Joseph A., Israel Gohberg and Leiba Rodman, Realization and interpolation of rational matrix functions, in: Operator theory: Advances and applications, Vol. 33 (Birkhauser Verlag, Basel). · Zbl 0708.15011
[2] Ball, Joseph A.; Gohlberg, Israel; Rodman, Leiba, Interpolation of Rational Matrix Functions (1990), Birkhauser Verlag: Birkhauser Verlag Basel, forthcoming · Zbl 0708.15011
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