Minimal ranks of some quaternion matrix expressions with applications

Abstract Suppose that p ( X ,  Y ) =  A  −  BX  −  X (∗) B (∗)  −  CYC (∗) and q ( X ,  Y ) =  A  −  BX  +  X (∗) B (∗)  −  CYC (∗) are quaternion matrix expressions, where A is persymmetric or perskew-symmetric. We in this paper derive the minimal rank formula of p ( X ,  Y ) with respect to pair of matrices X and Y  =  Y (∗) , and the minimal rank formula of q ( X ,  Y ) with respect to pair of matrices X and Y  = − Y (∗) . As applications, we establish some necessary and sufficient conditions for the existence of the general (persymmetric or perskew-symmetric) solutions to some well-known linear quaternion matrix equations. The expressions are also given for the corresponding general solutions of the matrix equations when the solvability conditions are satisfied. At the same time, some useful consequences are also developed.

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

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

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

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

[5]  Russell M. Reid,et al.  Classroom Note: Some Eigenvalue Properties of Persymmetric Matrices , 1997, SIAM Rev..

[6]  Guang-Jing Song,et al.  A new solvable condition for a pair of generalized Sylvester equations. , 2009 .

[7]  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..

[8]  Yongge Tian.  The Minimal Rank of the Matrix Expression $A - BX - YC$ , 2002 .

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

[10]  J. Woude,et al.  A system of real quaternion matrix equations with applications , 2009 .

[11]  Yongge Tian.  The solvability of two linear matrix equations , 2000 .

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

[13]  G. Styan,et al.  Equalities and Inequalities for Ranks of Matrices , 1974 .

[14]  Erik B. Dam,et al.  Quaternions, Interpolation and Animation , 2000 .

[15]  Ken Shoemake,et al.  Animating rotation with quaternion curves , 1985, SIGGRAPH.

[16]  A. Liao,et al.  On the existence of Hermitian positive definite solutions of the matrix equation Xs + A*X-tA = Q , 2008 .

[17]  Gabriele Nebe,et al.  Finite quaternionic matrix groups , 1998 .

[18]  Shao-Wen Yu,et al.  The Real Solutions to a System of Quaternion Matrix Equations with Applications , 2009 .

[19]  G. Dixon,et al.  On quaternions and octonions: Their geometry, arithmetic, and symmetry , 2004 .

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

[21]  Qing-Wen Wang,et al.  Ranks and the least-norm of the general solution to a system of quaternion matrix equations , 2009 .

[22]  Lawrence Charles Paulson,et al.  Quaternions , 1873, Nature.

[23]  Yongge Tian.  Ranks of Solutions of the Matrix Equation AXB = C , 2003 .

[24]  Yongge Tian,et al.  Extremal Ranks of Some Symmetric Matrix Expressions with Applications , 2006, SIAM J. Matrix Anal. Appl..

[25]  Peter Lancaster,et al.  Linear matrix equations from an inverse problem of vibration theory , 1996 .

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

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

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

[29]  Qingwen Wang,et al.  Extreme Ranks of Real Matrices in Solution of the Quaternion Matrix Equation AXB = C with Applications , 2010 .

[30]  Yuan Lei,et al.  Least squares Hermitian solution of the matrix equation (AXB, CXD) = (E, F) with the least norm over the skew field of quaternions , 2008, Math. Comput. Model..

[31]  U. Pinkall,et al.  Quaternions, Spinors, and Surfaces , 2002 .

[32]  William Rowan Hamilton,et al.  ON QUATERNIONS, OR ON A NEW SYSTEM OF IMAGINARIES IN ALGEBRA , 1847 .

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

[34]  Yongge Tian.  Completing triangular block matrices with maximal and minimal ranks , 2000 .

[35]  John H. Conway,et al.  On Quaternions and Octonions , 2003 .

[36]  Yongge Tian,et al.  The Maximal and Minimal Ranks of Some Expressions of Generalized Inverses of Matrices , 2002 .

[37]  S. Yau.  Mathematics and its applications , 2002 .

[38]  W. Hamilton.  ON QUATERNIONS By , 1999 .

[39]  Qing-Wen Wang,et al.  Ranks of the common solution to six quaternion matrix equations , 2011 .

[40]  F. Uhlig.  On the matrix equation AX=B with applications to the generators of a controllability matrix , 1987 .

[41]  Jack B. Kuipers,et al.  Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality , 2002 .

[42]  Sujit Kumar Mitra,et al.  Fixed rank solutions of linear matrix equations , 1972 .

[43]  J. P. Ward.  Quaternions and Cayley Numbers , 1997 .

[44]  Yongge Tian,et al.  The Minimum Rank of a 3 × 3 Partial Block Matrix , 2002 .

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

[46]  Dai Hua.  On the symmetric solutions of linear matrix equations , 1990 .

[47]  Guang-Jing Song,et al.  Maximal and Minimal Ranks of the Common Solution of Some Linear Matrix Equations over an Arbitrary Division Ring with Applications , 2009 .

[48]  Shao-Wen Yu,et al.  On solutions to the quaternion matrix equation AXB+CYD=E , 2008 .