Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications
Introduction
Extremal ranks of matrix expressions is one of important topics in matrix analysis and applications. In recent years, Tian has investigated extremal ranks of some matrix expressions over a field, and derived many rank formulas and their numerous consequences and applications. (see, e.g. [7], [8], [9], [10], [11], [12]).
Let the linear matrix expressionwhere A, A3, A4, B3, B4 are known matrices and X1, X2 are variant. In general, the matrix f(X1, X2) is variant for different choice of X1 and X2. Hence, the rank of f(X1, X2) may have different values with respect to the choice of X1 and X2. Because the rank of matrix is an integer between 0 and the minimum of row and column numbers of the matrix, extremal ranks, i.e., maximal and minimal ranks, of f(X1, X2) must exist with the variant matrices. Many problems in matrix theory and applications are closed related to extremal ranks of matrix expressions with variant entries. For example, a matrix equation A = A3X1 B3 + A4X2B4 is consistent if and only if the minimal rank of f(X1, X2) with respect to X1 and X2 are zero. In [6], [7], Tian has investigated the extremal ranks of f(X1, X2) over a field.
Uhlig in [1] presented the extremal ranks of solutions to the matrix equation AX = B. Mitra in [2] considered solutions with fixed ranks for the matrix equations AX = B and AXB = C. Tian in [3] has investigated the extremal rank solutions to the complex matrix equation AXB = C and gave some applications. Mitra [4] gave common solutions of minimal rank of the pair of complex matrix equations AX = C, XB = D, and studied the minimal ranks of common solutions to the pair of matrix equations A1XB1 = C1 and A2XB2 = C2 over a general field in [5].
The real quaternion matrices play a role in computer science, quantum physics, signal and color image processing, and so on (e.g. [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]).
Motivated by the work mentioned above and keeping some applications and interests of quaternion matrices in view, we in this paper investigate the extremal ranks of f(X1, X2) subject to consistent systems of linear matrix equations over and its applications. In Section 2, we derive the formulas of extremal ranks of f(X1, X2) subject to the consistent systems (1.2), (1.3)). As applications, we in Section 3 give necessary and sufficient conditions for the existence of general solutions to the systems of linear matrix equations over , the real quaternion field,andwhich has been studied by many authors (see, e.g. [13], [26], [27], [28], [29], [30], [31]). It is worthy to say that the system (1.4) can be used to investigate the centrosymmetric solution to the system (1.5), the symmetric solution to the systemover which can be used to consider the bisymmetric solution to the systemWe also give further research topics related to this paper in Section 4.
Throughout, we denote the set of all m × n matrices over the quaternion fieldby , the identity matrix with the appropriate size by I, the rank of a matrix A over by r(A), an inner inverse of a matrix A by A− which satisfies AA−A = A. Moreover, RA and LA stand for the two projectors LA = I − A−A, RA = I − AA− induced by A.
Section snippets
Main results
We begin with the following lemma which proof just like one over the complex field. Lemma 2.1 Let , be known and unknown. Then the following statements are equivalent: The system (1.2) is consistent.
In that case, the general solution of (1.2) iswhere Y1 is an arbitrary matrix over with appropriate dimension.
The following lemma is due to Marsglia and Styan [14], which can be
Solvable conditions to some systems of matrix equations
In this section, we use Theorem 2.5 to give necessary and sufficient conditions for the consistency to the systems (1.4), (1.5). Theorem 3.1 Let , , , , , , , , , , , , be known and , unknown. Then the system (1.4) is consistent if and only if the following equalities are all satisfied:
Conclusion
In this paper, we have derived the maximal and minimal ranks of the quaternion matrix expression A − A3X1B3 − A4X2B4 subject to consistent matrix equations A1X1 = C1, X1B1 = C2 and A2X2 = C3, X2B2 = C4. Using the results, we give some solvable conditions to some systems of quaternion matrix equations. Some known results can be viewed as the special cases of this paper. Similarly, we can study the maximal and minimal ranks of A − A3X1B3 − A4X2B4 subject to consistent matrix equations A1X1 = C1, A2X1 = C2 and X2B1 = C3
Acknowledgements
This research was supported by the Natural Science Foundation of China (No. 10471085), the Natural Science Foundation of Shanghai, the Development Foundation of Shanghai Educational Committee (No. 214498), and the Special Funds for Major Specialties of Shanghai Education Committee.
References (33)
On the matrix equation AX = B with applications to the generators of controllability matrix
Linear Algebra Appl.
(1987)A pair of simultaneous linear matrix equations A1XB1 = C1, A2XB2 = C2 and a programming problem
Linear Algebra Appl.
(1990)Upper and lower bounds for ranks of matrix expressions using generalized inverses
Linear Algebra Appl.
(2002)Completing block matrices with maximal and minimal ranks
Linear Algebra Appl.
(2000)More on maximal and minimal ranks of Schur complements with applications
Appl. Math. Comput.
(2004)Quaternions and matrices of quaternions
Linear Algebra Appl.
(1997)Right eigenvalues for quaternionic matrices: a topological approach
Linear Algebra Appl.
(1999)- et al.
Littlewood’s algorithm and quaternion matrices
Linear Algebra Appl.
(1999) - et al.
The spectral theorem in quaternions
Linear Algebra Appl.
(2003) - et al.
Singular value decomposition of matrices of quaternions: a new tool for vector-sensor signal processing
Signal Process.
(2004)
The general solution to a system of real quaternion matrix equations
Comput. Math. Appl.
Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations
Comput. Math. Appl.
Solving a kind of restricted matrix equations and Cramer rule
Appl. Math. Comput.
A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity
Linear Algebra Appl.
Fixed rank solutions of linear matrix equations
Sankhya Ser. A
Ranks of Solutions of the matrix equation AXB = C
Linear Multilinear Algebra
Cited by (44)
Characterization for the general solution to a system of matrix equations with quadruple variables
2014, Applied Mathematics and ComputationCitation Excerpt :Hence, the system (1.5) has a solution. The solvability condition (c) is Theorem 3.1 in [12]. Wang et al. in [21] gave an expression of the general solution to the system (2.31).
Equality of the BLUPs under the mixed linear model when random components and errors are correlated
2013, Journal of Multivariate AnalysisThe common bisymmetric nonnegative definite solutions with extreme ranks and inertias to a pair of matrix equations
2011, Applied Mathematics and ComputationCitation Excerpt :As we known, ranks of solutions of linear matrix equations have been considered previously by several authors (see, e.g. [11,15–31]).
The (P, Q)-(skew)symmetric extremal rank solutions to a system of quaternion matrix equations
2011, Applied Mathematics and ComputationMinimal ranks of some quaternion matrix expressions with applications
2010, Applied Mathematics and Computation