Solution Of Matrix Equation AXB Research Articles

ADMM-Based Methods for Nearness SkewSymmetric and Symmetric Solutions of Matrix Equation AXB = C

ADMM-Based Methods for Nearness SkewSymmetric and Symmetric Solutions of Matrix Equation AXB = C

On solutions of matrix equation AXB=C under semi-tensor product

ABSTRACT The semi-tensor product, which was initially proposed by Cheng et al. [An introduction to semi-tensor product of matrices and its applications. World Scientific; 2012], has been extensively applied in Boolean control networks, graph colouring, game theory, cryptographic algorithms and so on. In this article, motivated by the existing work by Yao et al. [J Franklin Inst. 2016;353:1109–1131], we further investigate the solvability of the matrix equation AXB=C with respect to semi-tensor product. The case of matrix-vector equation, in which the required unknown X be a vector, is studied first. Compatible condition for matrix dimensions, necessary and sufficient conditions and concrete solving methods are established. Based on this, the solvability of the matrix equation case, in which the unknown X be a matrix, under semi-tensor product is then studied. For each part, several elementary examples are presented to illustrate the efficiency of the results.

Conjugate Gradient Algorithm for the Symmetric Arrowhead Solution of Matrix Equation AXB=C

Conjugate Gradient Algorithm for the Symmetric Arrowhead Solution of Matrix Equation AXB=C

Conjugate Gradient Algorithm for the Symmetric Arrowhead Solution of Matrix Equation AXB=C

Conjugate Gradient Algorithm for the Symmetric Arrowhead Solution of Matrix Equation AXB=C

Optimization problems of the rank and inertia corresponding to a Hermitian least-squares problem

Generally, the least-squares problem can be solved by the normal equation. Based on the projection theorem, we propose a direct method to investigate the maximal and minimal ranks and inertias of the least-squares solutions of matrix equation AXB = C under Hermitian constraint, and the corresponding formulas for calculating the rank and inertia are derived.

The {P,Q,k+1}-reflexive solution of matrix equation AXB=C

Let P∈ℂ m×m and Q∈ℂ n×n be Hermitian and {k+1}-potent matrices, i.e., P k+1=P=P ∗, Q k+1=Q=Q ∗, where (⋅)∗ stands for the conjugate transpose of a matrix. A matrix X∈ℂ m×n is called {P,Q,k+1}-reflexive (anti-reflexive) if PXQ=X (PXQ=−X). In this paper, the matrix equation AXB=C subject to {P,Q,k+1} reflexive and anti-reflexive constraints are studied by converting into two simpler cases: k=1 and k=2. The solvability conditions and the general solution are obtained, and then the maximal and minimal ranks among the solutions are derived. Finally, the associated optimal approximation problem for a given matrix is considered.

A matrix LSQR iterative method to solve matrix equation AXB=C

This paper is a matrix iterative method presented to compute the solutions of the matrix equation, AXB=C, with unknown matrix X∈𝒮, where 𝒮 is the constrained matrices set like symmetric, symmetric-R-symmetric and (R, S)-symmetric. By this iterative method, for any initial matrix X 0∈𝒮, a solution X* can be obtained within finite iteration steps if exact arithmetics were used, and the solution X* with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. The solution , which is nearest to a given matrix ˜X in Frobenius norm, can be obtained by first finding the minimum Frobenius norm solution of a new compatible matrix equation. The numerical examples given here show that the iterative method proposed in this paper has faster convergence and higher accuracy than the iterative methods proposed in [G.-X. Huang, F. Yin, and K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C , J. Comput. Appl. Math. 212 (2008), pp. 231–244; Y. Lei and A.-P. Liao, A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetric matrices, Appl. Math. Comput. 188 (2007), pp. 499–513; Z.-Y. Peng, An iterative method for the least squares symmetric solution of the linear matrix equation AXB=C , Appl. Math. Comput. 170 (2005), pp. 711–723; Y.-X. Peng, X.-Y. Hu, and L. Zhang, An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C , Appl. Math. Comput. 160 (2005), pp. 763–777].

On additive decompositions of solutions of the matrix equation AXB=C

This paper considers decompositions of solutions of the linear matrix equation AXB=C into the sums of solutions of other two linear matrix equations A 1 X 1 B 1=C 1 and A 2 X 2 B 2=C 2. Some applications are also given on additive decompositions of generalized inverses, as well as decompositions of solutions of matrix equations into the sums of solutions of their small equations.

Ranks of Solutions of the Matrix Equation AXB = C

The purpose of this article is to solve two problems related to solutions of a consistent complex matrix equation AXB = C : (I) the maximal and minimal ranks of solution to AXB = C , and (II) the maximal and minimal ranks of two real matrices X 0 and X 1 in solution X = X 0 + iX 1 to AXB = C . As applications, the maximal and minimal ranks of two real matrices C and D in generalized inverse (A + iB)- = C + iD of a complex matrix A + iB are also examined.

On solutions of matrix equation AXB + CYD = F

In this paper, the matrix equation with two unknown matrices X, Y of form AXB + CYD = F is discussed. By applying the canonical correlation decomposition (CCD) of matrix pairs, we obtain expressions of the least-squares solutions of the matrix equation, and sufficient and necessary conditions for the existence and uniqueness of the solutions. We also derive a general form of the solutions. We also study the least-squares Hermitian (skew-Hermitian) solutions of equation AXA H + CYC H = F.

Classroom Note:Centrosymmetric Matrices

Attention is drawn to some earlier work on the odd--even properties of eigenvectors of centrosymmetric matrices.