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Analytical best approximate Hermitian and generalized skew-Hamiltonian solution of matrix equation \(AXA^{\mathrm{H}}+CYC^{\mathrm{H}}=F\). (English) Zbl 1417.65125

Summary: In this paper, we consider the matrix equation \(AXA^{\mathrm{H}}+CYC^{\mathrm{H}}=F\), where \(A\), \(C\) and \(F\) are given matrices with appropriate sizes and \([X,Y]\) is an unknown Hermitian and generalized skew-Hamiltonian matrix pair. Based on matrix differential calculus and projection theorem in inner product spaces, we exploit the best approximate solution \([\hat{X},\hat{Y}]\) in the set \(\mathcal{S}\) to a given matrix pair \([X^\ast,Y^\ast]\), where \(\mathcal{S}\) signifies the least-squares Hermitian and generalized skew-Hamiltonian solution set of the matrix equation \(AXA^{\mathrm{H}}+CYC^{\mathrm{H}}=F\). The analytical expression of the best approximate solution is presented by applying the canonical correlation decomposition and the generalized singular value decomposition. Finally, a numerical algorithm and an illustrated example are given.

MSC:

65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities
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