Matrix Equation AX Research Articles

The *Congruence Class of the Solutions to a System of Matrix Equations

We present the *congruence class of the least-square and the minimum norm least-square solutions to the system of complex matrix equationAX=C, XB=Dby generalized singular value decomposition and canonical correlation decomposition.

The Solutions to Matrix EquationAX=Bwith Some Constraints

LetPbe a given Hermitian matrix satisfyingP2=I. Using the eigenvalue decomposition ofP, we consider the least squares solutions to the matrix equationAX=Bwith the constraintsPX=XPandX*=X. A similar problem of this matrix equation with generalized constrained is also discussed.

(Anti-)Hermitian Generalized (Anti-)Hamiltonian Solution to a System of Matrix Equations

We mainly solve three problems. Firstly, by the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the (anti-)Hermitian generalized (anti-)Hamiltonian solutions to the system of matrix equationsAX=B,XC=Dare derived, respectively. Secondly, the optimal approximation solutionmin⁡X∈K⁡∥X^-X∥is obtained, whereKis the (anti-)Hermitian generalized (anti-)Hamiltonian solution set of the above system andX^is the given matrix. Thirdly, the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solutions are considered. In addition, algorithms about computing the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solution and the corresponding numerical examples are presented.

The solution of the equation AX + BX ⋆ = 0

We give a complete solution of the matrix equation AX + BX ⋆ = 0, where A, B ∈ ℂ m×n are two given matrices, X ∈ ℂ n×n is an unknown matrix, and ⋆ denotes the transpose or the conjugate transpose. We provide a closed formula for the dimension of the solution space of the equation in terms of the Kronecker canonical form of the matrix pencil A + λB, and we also provide an expression for the solution X in terms of this canonical form, together with two invertible matrices leading A + λB to the canonical form by strict equivalence.

On the structure of the spectral solutions of the Yang–Baxter matrix equation

Let A be a square matrix. We investigate the structure of the spectral solutions of the nonlinear matrix equation AXA=XAX by showing that any semisimple eigenvalue of A with multiplicity at least 2 gives rise to infinitely many solutions. Such solutions can be obtained by means of the projections onto subspaces of the corresponding eigenspace.

Further Solutions of a Yang-Baxter-like Matrix Equation

AbstractThe Yang-Baxter-like matrix equation AXA = XAX is reconsidered, and an infinite number of solutions that commute with any given complex square matrix A are found. Our results here are based on the fact that the matrix A can be replaced with its Jordan canonical form. We also discuss the explicit structure of the solutions obtained.

A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations

In this paper, we discuss the properties of the eigenvalues related to the symmetric positive definite matrices. Several new results are established to express the structures and bounds of the eigenvalues. Using these results, a family of iterative algorithms are presented for the matrix equation AX=F and the coupled Sylvester matrix equations. The analysis shows that the iterative solutions given by the least squares based iterative algorithms converge to their true values for any initial conditions. The effectiveness of the proposed iterative algorithm is illustrated by a numerical example.

On solutions of the quaternion matrix equation [formula omitted] and their applications in color image restoration

By using the complex representation of quaternion matrices, and the Moore–Penrose generalized inverse, we derive the expressions of the least squares solution with the least norm, the least squares pure imaginary solution with the least norm, and the least squares real solution with the least norm for the quaternion matrix equation AX=B, respectively. Finally, we discuss their applications in color image restoration.

Numerical algorithms for solving matrix equations AX + BX T = C and AX + BX* = C

Conditions for the unique solvability of matrix equations of the form AX + BX T = X and AX + BX* = C are found. Numerical algorithms of the Bartels-Stewart type for solving such equations are described. Certain numerical tests with these algorithms are presented. In particular, the situation where the conditions for unique solvability are “almost” violated is modeled, and the deterioration of the quality of the computed solution in this situation is traced through.

THE EXTREMAL RANKS AND INERTIAS OF THE LEAST SQUARES SOLUTIONS TO MATRIX EQUATION AX = B SUBJECT TO HERMITIAN CONSTRAINT

In this paper, the formulas for calculating the extremal ranks and inertias of the Hermitian least squares solutions to matrix equation AX = B are established. In particular, the necessary and sufficient conditions for the existences of the positive and nonnegative definite solutions to this matrix equation are given. Meanwhile, the least squares problem of the above matrix equation with Hermitian R-symmetric and R-skew symmetric constraints are also investigated.

On the generalized bi (skew-) symmetric solutions of a linear matrix equation and its procrust problems

In this paper, the solvability conditions and the explicit expressions of the generalized bisymmetric and bi-skew-symmetric solutions of the matrix equation AX=B are respectively established by applying two methods. Then the maximal and minimal ranks of the solutions are derived. If the solvability conditions are not satisfied, the generalized bisymmetric and bi-skew-symmetric least squares solutions of the matrix equation are considered, and the generalized bisymmetric and bi-skew-symmetric least squares solutions with the minimum norm are also obtained. In addition, two algorithms are provided to compute the generalized bi (skew-) symmetric least squares solution, and some examples are given to illustrate that the algorithms are feasible.

Numerical solution of the matrix equations AX + X T B=C and X + AX T B=C with rectangular coefficients A and B

It is shown how one should supplement the orthogonal algorithms previously proposed by the authors for the equations in the title of the article with square matrix coefficients for these algorithms to be applicable to equations with rectangular coefficients, provided that the latter satisfy conditions of unique solvability. Bibliography: 3 titles.

The matrix equations AX + BX T = C and AX + BX* = C

The matrix equations AX + BX T = C and AX + BX* = C

The symmetric solutions of the matrix inequality AX≥B in least-squares sense

In this paper, we present an iterative method to compute the symmetric solutions of the matrix inequality AX≥B in least-squares sense. The presented iteration method can be also used to compute the symmetric solutions of the matrix equation AX=B and the consistent matrix inequality AX≥B. Numerical experiments are given to illustrate the usefulness of the proposed approach.

Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations

The Drazin inverse solutions of the matrix equations AX=B,XA=B and AXB=D are considered in this paper. We use both the determinantal representations of the Drazin inverse obtained earlier by the author and in the paper. We get analogs of the Cramer rule for the Drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations, X′+AX=B and X′+XA=B, where the matrix A is singular.

Numerical algorithm for solving the matrix equation AX + X* B = C

An algorithm of the Bartels-Stewart type for solving the matrix equation AX + X*B = C is suggested. By applying the QZ-algorithm to the original equation, it is transformed into an equation of the same type with triangular matrix coefficients A and B. The resulting matrix equation is equivalent to the sequence of a system of linear equations with a smaller order of the coefficients of the desired solution. Using numerical examples, the authors simulate a situation where the conditions of a unique solution are “almost” violated. Deterioration of the calculated solutions is in this case followed.

Approximate Solution of LR Fuzzy Sylvester Matrix Equations

The fuzzy Sylvester matrix equationAX~+X~B=C~in whichA,Barem×mandn×ncrisp matrices, respectively, andC~is anm×nLR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Some examples are given to illustrate the proposed method.

Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations AX = B and AXA * = B

This paper studies algebraic properties of Hermitian solutions and Hermitian definite solutions of the two types of matrix equations AX = B and AXA * = B. We first establish a variety of rank and inertia formulas for calculating the maximal and minimal ranks and inertias of Hermitian solutions and Hermitian definite solutions of the matrix equations AX = B and AXA * = B, and then use them to characterize many qualities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations and their variations.

Ranks of submatrices in (skew-)Hermitian solutions to a quaternion matrix equation

Assume that X and Y are Hermitian solutions to quaternion matrix equation AXA ∗+BYB ∗=C, which are partitioned into 2×2 block forms. We in this paper give the maximal and minimal ranks of submatrices in the Hermitian solutions X=X ∗, Y=Y ∗, and establish necessary and sufficient conditions for the submatrices to be zero, unique as well as independent. The findings of this paper widely extend the known results in the literature.

A Nontrivial Solution to a Stochastic Matrix Equation

Abstract.If A is a nonsingular matrix such that its inverse is a stochastic matrix, the classic Brouwer fixed point theorem implies that the matrix equation AXA = XAX has a nontrivial solution. An explicit expression of this nontrivial solution is found via the mean ergodic theorem, and fixed point iteration is considered to find a nontrivial solution.