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A simultaneous decomposition of a matrix triplet with applications. (English) Zbl 1249.15020
The paper concentrates on a matrix expression $$A-BXB^{*}-CYC^{*}$$, where $$A$$ is a given square Hermitian or skew-Hermitian matrix, $$B, C$$ are given (generally rectangular) matrices and $$X, Y$$ are variable matrices. First, motivated by the generalized singular value decomposition of two matrices a simultaneous decomposition of the matrix triplet $$(A,B,C)$$ is derived. Using this result, closed formulas for the maximal and minimal possible ranks of the matrix $$A-BXB^{*}-CYC^{*}$$ are given under the assumption $$X = \pm X^{*}, Y = \pm Y^{*}$$. Moreover, necessary and sufficient conditions for existence of two Hermitian or skew-Hermitian solutions of the equation $$A=BXB^{*}+CYC^{*}$$ are derived. These conditions can be simply verified and involve ranks of the individual matrices $$A,B,C$$. Formulas for the solutions are also presented. Finally, some further open problems related to the considered expression $$A-BXB^{*}-CYC^{*}$$ are summarized.

##### MSC:
 15A24 Matrix equations and identities 15A23 Factorization of matrices 15A03 Vector spaces, linear dependence, rank, lineability 15B57 Hermitian, skew-Hermitian, and related matrices
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