Quaternion generalized singular value decomposition and its applications. (English) Zbl 1093.15014

The authors derive a theorem of the generalized singular value decomposition of quaternion matrices(QGSVD), study the solution of general quaternion matrix equation \(\mathcal{AXB-CYD=E}\), and obtain a quaternionic Roth’s theorem. This paper also suggests sufficient and necessary conditions for the existence and uniqueness of solutions and explicit forms of the solutions of the equation.


15A24 Matrix equations and identities
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A18 Eigenvalues, singular values, and eigenvectors
Full Text: DOI


[1] Xu Guiping, Wei Musheng, Zheng Daosheng. On solution of matrix equation AXB+CYD=E, Linear Algebra Appl, 1998, 279:93–109. · Zbl 0933.15024
[2] Chu K E. Singular value and generalized singular value decompositions and the solution of linear matrix equtions, Linear Algebra Appl, 1987, 87:83–98. · Zbl 0612.15003
[3] Bunse-Gerstner A, Byers R, Mehrmann V. A quaternion QR algorithm, Numer Math, 1989, 55: 83–95. · Zbl 0681.65024
[4] Jiang T, Chen L. Generalized diagonalization of matrices over quaternion field, Applied Mathematics and Mechanics, 1999, 20(11):1230–1210. · Zbl 0946.35085
[5] Zhang Fuzhen. Quaternions and matrices of quaternions, Linear Algebra Appl, 1997, 251:21–57. · Zbl 0873.15008
[6] Roth W E. The equations AXB=C and AXB=C in matrices, Proc Amer Math Soc, 1952, 3:392–396. · Zbl 0047.01901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.