Solvability of a quaternion matrix equation. (English) Zbl 1020.15014
The author studies the solvability of the matrix equations \((*)\) \(AXB=D\) and \((**)\) \(A^*X^*B^*\pm BXA=D\) where \(A\), \(B\), \(D\) are quaternion matrices. He gives necessary and sufficient conditions for the solvability of \((*)\) (resp. of \((**)\)) in Hermitian and in skew-Hermitian matrices (resp. in quaternion matrices), and he presents formulas for the general solutions when the conditions are met. These formulas involve the matrices \(A\), \(B\), \(D\) and their Moore-Penrose inverses. (The Moore-Penrose inverse \(A^+\) of the \(m\times n\) quaternion matrix \(A\) is an \(n\times m\)-matrix \(X\) satisfying the conditions \(AXA=A\), \(XAX=X\), \((AX)^*=AX\), \((XA)^*=XA\).).
Reviewer: Vladimir P.Kostov (Nice)
MSC:
| 15A24 | Matrix equations and identities |
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
Keywords:
quaternion matrix; (skew)-Hermitian matrix; (quaternion) matrix equation; Moore-Penrose inverse; general solutionsReferences:
| [1] | Chu, K. W. E., Symmetric solutions of linear matrix equations by matrix decompositions, Linear Algebra and Its Application, 119, 35-50 (1989) · Zbl 0688.15003 · doi:10.1016/0024-3795(89)90067-0 |
| [2] | Dai, Hua, On the symmetric solutions of linear matrix equations, Linear Algebra and Its Application, 131, 1-7 (1990) · Zbl 0712.15009 · doi:10.1016/0024-3795(90)90370-R |
| [3] | Don, F. J. H., On the symmetric solutions of linear matrix equation, Linear Algebra and Its Application, 93, 1-7 (1987) · Zbl 0622.15001 · doi:10.1016/S0024-3795(87)90308-9 |
| [4] | Huang, Liping, Transformation theroems of quaternion matrics and their applications, Northeaster Mathematical, 9, 1, 69-80 (1993) · Zbl 0794.15009 |
| [5] | Huang, Liping, The matrix equation AXB−GXD=E over the quaternion field, Linear Algebra and Its Application, 234, 197-208 (1996) · Zbl 0840.15017 · doi:10.1016/0024-3795(94)00103-0 |
| [6] | Huang, Liping, The solvability of matrix equation AXB+CYD=E over a ring, Advances in Mathematics, 26, 3, 269-275 (1997) · Zbl 0886.15011 |
| [7] | Huang, Liping, The quaternion matrix equation Σ A^iXB_i=E, Acta Mathematica Sinica, New Series, 14, 1, 91-98 (1998) · Zbl 0908.15007 · doi:10.1007/BF02563888 |
| [8] | Zhuang, Wajin, The quaternion matrix equation, Acta Mathematica Sinica, 30, 5, 688-694 (1987) · Zbl 0636.15008 |
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