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Ranks of the common solution to six quaternion matrix equations. (English) Zbl 1304.15006

Summary: A new expression is established for the common solution to six classical linear quaternion matrix equations \(A_1X=C_1\), \(XB_1=C_3\), \(A_2X=C_2\), \(XB_2=C_4\), \(A_3XB_3=C_5\), \(A_4XB_4=C_6\) which was investigated recently by Q.-W. Wang et al. [Appl. Math. Comput. 195, No. 2, 721–732 (2008; Zbl 1149.15011)]. Formulas are derived for the maximal and minimal ranks of the common solution to this system. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition is presented for the invariance of the rank of the general solution to this system. Some known results can be regarded as the special cases of the results in this paper.

MSC:

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

Citations:

Zbl 1149.15011
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Full Text: DOI

References:

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