对于四元数矩阵方程组AXAη∗+ BYBη∗= E, CYCη∗+ DZDη∗= F , 首先运用 4 个矩阵的奇异值分解, 给出四元数矩阵方程组有η-Hermitian解的充要条件; 然后, 利用该充要条件给出矩阵方程组η-Hermitian解的表达式.
This paper derives a necessary and sufficient condition for the existence of the η-Hermitian solutions to the quaternion matrix equationsAXAη∗+ BYBη∗= E, CYCη∗+ DZDη∗= F by using the singular value decomposition of four matrices, and gives an expression of the η-Hermitian solutions to the quaternion matrix equations.
[1] Ell T, Sangwine S J. Quaternion involutions and anti-involutions [J]. Comput Math Appl, 2007, 53(1): 137-143.
[2] Horn R A, Zhang F. A generalization of the complex Autonne-Takagi factorization to quaternion matrices [J]. Linear Multilinear Algebra, 2012, doi: 10.1080/03081087.2011.618838.
[3] Wang Q W, van der Woude J W, Yu S W. An equivalence canonical form of a matrix triplet over an arbitrary division ring with applications [J]. Sci China Math, 2011, 54(5): 907-924.
[4] Wang Q W. A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity [J]. Linear Algebra Appl, 2004, 384: 43-54.
[5] Yuan S F, Wang Q W. Two special kinds of least squares solutions for the quaternion matrix equation AXB + CYD = E [J]. Electron J Linear Algebra, 2012, 23: 257-274.
[6] He Z H, Wang Q W. A real quaternion matrix equation with applications [J]. Linear Multilinear Algebra, 2012, doi: 10.1080/03081087.2012.703192.
[7] Chu D L, De Moor B. On a variational formulation of the QSVD and the RSVD [J]. Linear Algebra Appl, 2000, 311: 61-78.
[8] Liu Y H, Tian Y G. A simultaneous decomposition of a matrix triplet with applications [J]. Numer Linear Algebra Appl, 2011, 18(1): 69-85.
[9] de Moor B, Zha H Y. A tree of generalization of the ordinary singular value decomposition [J]. Linear Algebra Appl, 1991, 147: 469-500.
[10] Paige C C, Saunders M A. Towards a generalized singular value decomposition [J]. SIAM J Numer Anal, 1981, 18(3): 398-405.
[11] van Loan C F. Generalizing the singular value decomposition [J]. SIAM J Numer Anal, 1976, 13(1): 76-83.
[12] Wang Q W, Zhang X, Van Der Woude J W. A new simultaneous decomposition of a matrix quaternity over an arbitrary division ring with applications [J]. Comm Algebra, 2012, 40: 2309-2342.
[13] Zha H Y. The restricted singular value decomposition of matrix triplets [J]. SIAM J Matrix Anal Appl, 1991, 12(1): 172-194.
[14] Zha H Y. The product-product singular value decomposition of matrix triplets [J]. BIT, 1991, 31(4): 711-726.
