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Solving and algorithm to system of quaternion Sylvester-type matrix equations with \(\ast\)-Hermicity. (English) Zbl 1501.15010

The authors consider a type of Sylvester-like (systems of) matrix equations, namely, the following system of Hermitian Sylvester matrix equations: \begin{align*} A_1 X_1 A_1^* + B_1 Y_1 B_1^* = C_1 , \quad C_1 = C_1^* , \\ A_2 X_2 A_2^* + B_2 Y_1 B_2^* = C_2 , \quad C_2 = C_2^* , \end{align*} over the quaternion algebra \(\mathbb H\). They express the Hermitian solutions to this system in terms of the Moore-Penrose (MP) inverses, and give an algorithm for finding these solutions by using determinantal representations of the quaternionic MP-inverse obtained earlier by the second author in [Abstr. Appl. Anal. 2019, Article ID 5926832, 14 p. (2019; Zbl 1474.15041)]. A closed form formula expressing the general solution is established when solvability conditions are satisfied. Finally, a numerical example is given to illustrate the authors’ results.

MSC:

15A24 Matrix equations and identities
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A09 Theory of matrix inversion and generalized inverses
15A10 Applications of generalized inverses

Citations:

Zbl 1474.15041
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References:

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