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A system of Sylvester-type quaternion matrix equations with ten variables. (English) Zbl 1497.15020

Summary: This paper studies a system of three Sylvester-type quaternion matrix equations with ten variables \(A_i X_i + Y_i B_i + C_i Z_i D_i + F_i Z_{i+1} G_i = E_i,\; i = \overline{1,3}\). We derive some necessary and sufficient conditions for the existence of a solution to this system in terms of ranks and Moore-Penrose inverses of the matrices involved. We present the general solution to the system when the solvability conditions are satisfied. As applications of this system, we provide some solvability conditions and general solutions to some systems of quaternion matrix equations involving \(\phi\)-Hermicity. Moreover, we give some numerical examples to illustrate our results. The findings of this paper extend some known results in the literature.

MSC:

15A24 Matrix equations and identities
15A09 Theory of matrix inversion and generalized inverses
15A03 Vector spaces, linear dependence, rank, lineability
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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