Using the GSVD and the lifting technique to find {P, k+1} reflexive and anti-reflexive solutions of AXB=C

Abstract The generalized singular value decomposition (GSVD) and the lifting technique combined with the Kronecker product are exploited to find reflexive and anti-reflexive (with respect to a generalized { k + 1 } -reflection matrix P ) solutions of the matrix equation A X B = C . The computational cost of the presented algorithm is studied and several numerical examples are presented.

[1]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[2]  H. Zha.  The restricted singular value decomposition of matrix triplets , 1991 .

[3]  D S Cvetkovic Ilic.  THE REFLEXIVE SOLUTIONS OF THE MATRIX EQUATION AXB=C , 2006 .

[4]  Yimin Wei,et al.  An improvement on the perturbation of the group inverse and oblique projection , 2001 .

[5]  Néstor Thome,et al.  {k}-Group Periodic Matrices , 2006, SIAM J. Matrix Anal. Appl..

[6]  H. Wimmer.  Decomposition and Parametrization of Semidefinite Solutions of the Continuous-Time Algebraic Riccati Equation , 1994 .

[7]  D. S. Cvetkovi-Iliíc.  The reflexive solutions of the matrix equation AX B = C , 2006 .

[8]  Alicia Herrero,et al.  An algorithm to check the nonnegativity of singular systems , 2007, Appl. Math. Comput..

[9]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[10]  Alicia Herrero,et al.  Nonnegativity, stability, and regularization of discrete-time descriptor systems☆ , 2010 .

[11]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[12]  Lei Zhang,et al.  The reflexive and anti-reflexive solutions of the matrix equation AHXB=C , 2007 .

[13]  Alan L. Andrew.  Classroom Note: Centrosymmetric Matrices , 1998, SIAM Rev..

[14]  Xi-Yan Hu,et al.  The reflexive and anti-reflexive solutions of the matrix equation AX = B , 2003 .

[15]  Qing-Wen Wang,et al.  Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra , 2002 .

[16]  Yongge Tian,et al.  How to use RSVD to solve the matrix equation A=BXC , 2010 .