Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications

Abstract In this paper, we establish the formulas of the maximal and minimal ranks of the quaternion matrix expression C 4 - A 4 XB 4 where X is a variant quaternion matrix subject to quaternion matrix equations A 1 X = C 1 , XB 2 = C 2 , A 3 XB 3 = C 3 . As applications, we give a new necessary and sufficient condition for the existence of solutions to the system of matrix equations A 1 X = C 1 , XB 2 = C 2 , A 3 XB 3 = C 3 , A 4 XB 4 = C 4 , which was investigated by Wang [Q.W. Wang, A system of four matrix equations over von Neumann regular rings and its applications, Acta Math. Sin., 21(2) (2005) 323–334], by rank equalities. In addition, extremal ranks of the generalized Schur complement D - CA - B with respect to an inner inverse A − of A , which is a common solution to quaternion matrix equations A 1 X = C 1 , XB 2 = C 2 , are also considered. Some previous known results can be viewed as special cases of the results of this paper.

[1]  Yongge Tian.  Upper and lower bounds for ranks of matrix expressions using generalized inverses , 2002 .

[2]  Sujit Kumar Mitra,et al.  A pair of simultaneous linear matrix equations A1XB1 = C1, A2XB2 = C2 and a matrix programming problem , 1990 .

[3]  Qing-Wen Wang,et al.  Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application , 2007, Appl. Math. Comput..

[4]  T. Andô.  Generalized Schur complements , 1979 .

[5]  Miroslav Fiedler,et al.  Remarks on the Schur complement , 1981 .

[6]  Qing-Wen Wang,et al.  Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations , 2005 .

[7]  Fuzhen Zhang.  Quaternions and matrices of quaternions , 1997 .

[8]  D. Farenick,et al.  The spectral theorem in quaternions , 2003 .

[9]  S. Mitra.  The matrix equations AX = C, XB = D , 1984 .

[10]  C. Eddie Moxey,et al.  Hypercomplex correlation techniques for vector images , 2003, IEEE Trans. Signal Process..

[11]  Stephen J. Sangwine,et al.  Color image decomposition using quaternion singular value decomposition , 2003 .

[12]  Qingwen Wang,et al.  A System of Matrix Equations and a Linear Matrix Equation Over Arbitrary Regular Rings with Identity , 2003 .

[13]  D. Carlson.  What are Schur complements, anyway? , 1986 .

[14]  S. Adler,et al.  Quaternionic quantum mechanics and quantum fields , 1995 .

[15]  Andrew Baker,et al.  Right eigenvalues for quaternionic matrices: A topological approach , 1999 .

[16]  Nicolas Le Bihan,et al.  Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing , 2004, Signal Process..

[17]  Qingwen Wang.  The general solution to a system of real quaternion matrix equations , 2005 .

[18]  T. Markham,et al.  A Generalization of the Schur Complement by Means of the Moore–Penrose Inverse , 1974 .

[19]  Qing-Wen Wang,et al.  Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications , 2006, Appl. Math. Comput..

[20]  R. Tennant.  Algebra , 1941, Nature.

[21]  Nicolas Le Bihan,et al.  Quaternion principal component analysis of color images , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[22]  Qing-Wen Wang,et al.  A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications , 2005 .

[23]  Yongge Tian,et al.  The maximal and minimal ranks of A − BXC with applications , 2003 .

[24]  Dennis I. Merino,et al.  Littlewood's algorithm and quaternion matrices , 2007, 0709.2466.

[25]  S. Leo,et al.  Right eigenvalue equation in quaternionic quantum mechanics , 2000, math-ph/0002051.

[26]  Yongge Tian,et al.  More on maximal and minimal ranks of Schur complements with applications , 2004, Appl. Math. Comput..

[27]  Nicolas Le Bihan,et al.  Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion Householder transformations , 2006, Appl. Math. Comput..