Symmetric matrix polynomial equations. (English) Zbl 0599.93008

The following matrix polynomial equation is studied (1) \(A^*X+X^*A=2B\), where A,B,X are real \(n\times n\) matrix polynomials, and \(A^*(s)=A(-s)\). Here A and B are given and X is unknown. It is assumed that A is stable, i.e. det A has no zeros in the closed right half-plane, and that \(XA^{-1}\) is proper. A general form of solutions is given, and particular attention is paid to the case when B(s) is positive definite for pure imaginary s. The discrete analogue of (1) is considered as well (in this case \(A^*(s)=A(s^{-1})\), and instead of being polynomials the matrix functions A,B,X are of type \(\sum^{p}_{j=-p}s^ jA_ j\), where \(A_ j\) are \(n\times n\) matrices). Numerical algorithms for solving (1) and its discrete counterparts are devised and tested.
Reviewer: L.Rodman


93B25 Algebraic methods
15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
93B40 Computational methods in systems theory (MSC2010)
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