A numerical algorithm for solving the matrix equation AX + XTB = C1

An algorithm of the Bartels-Stewart type for solving the matrix equation AX + X ^T B = C is proposed. By applying the QZ algorithm, the original equation is reduced to an equation of the same type having triangular matrix coefficients A and B. The resulting matrix equation is equivalent to a sequence of low-order systems of linear equations for the entries of the desired solution. Through numerical experiments, the situation where the conditions for unique solvability are “nearly” violated is simulated. The loss of the quality of the computed solution in this situation is analyzed.     

On the calculation of neutral subspaces of a matrix

A technique for constructing solutions to the quadratic matrix equation X ^T DX +AX + X ^T B + C = 0 is outlined. It is similar to the well-known Schur approach for solving algebraic Riccati equations.     

The Constrained Solutions of Two Matrix Equations

We study the symmetric positive semidefinite solution of the matrix equation AX ₁ A ^T + BX ₂ B ^T = C, where A is a given real m×n matrix, B is a given real m×p matrix, and C is a given real m×m matric, with m, n, p positive integers; and the bisymmetric positive semidefinite solution of the matrix equation D ^T XD = C, where D is a given real n×m matrix, C is a given real m×m matrix, with m, n positive integers. By making use of the generalized singular value decomposition, we derive general analytic formulae, and present necessary and sufficient conditions for guaranteeing the existence of these solutions. Received December 17, 1999, Revised January 10, 2001, Accepted March 5, 2001     

Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations

The consistent conditions and the general expressions about the Hermitian solutions of the linear matrix equations AXB=C and (AX, XB)=(C, D) are studied in depth, where A, B, C and D are given matrices of suitable sizes. The Hermitian minimum F‐norm solutions are obtained for the matrix equations AXB=C and (AX, XB)=(C, D) by Moore–Penrose generalized inverse, respectively. For both matrix equations, we design iterative methods according to the fundamental idea of the classical conjugate direction method for the standard system of linear equations. Numerical results show that these iterative methods are feasible and effective in actual computations of the solutions of the above‐mentioned two matrix equations. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical solution of matrix equations of the form X + AX T B = C

A review of numerical methods for solving matrix equations of the form X + AX ^T B = C is given. The methods under consideration were implemented in the Matlab environment. The performances of these methods are compared, including the case where the conditions for unique solvability are “almost” violated.     

On the numerical solution ofAX −XB =C

In this note we analyze the numerical solution ofAX –XB =C when a Galerkin method is applied, assuming thatB has much smaller size thanA. We show that the corresponding Galerkin equation can be obtained from the truncation of the original problem, if matrix polynomials are used for writing the analytical solutionX. Moreover, we provide some relations between the separation ofA andB in their natural space and that in the projected space. Experimental tests validate some of the theoretical results and show the rate of applicability of the method with respect to a standard linear system solver.Part of this work was done while the author was at the Dipartimento di Fisica, Universitá di Bologna, Italy.     

Roth's theorems over commutative rings

In 1952, W.E. Roth showed that matrix equations of the forms AX?YB = C and AX?XB = C over fields can be solved if and only if certain block matrices built from A, B, and C are equivalent or similar. We show here that these criteria remain valid over arbitrary commutative rings. To do this, we use standard commutative algebra methods to reduce to the case of Artinian rings, where a simple argument with     

Numerical solution of the matrix equations AX + X T B = C and AX + X*B = C in the self-adjoint case

The numerical algorithms for solving equations of the type AX + X ^T B = C or AX + X*B = C that were earlier proposed by the authors are now modified for the situations where these equations can be regarded as self-adjoint ones. The economy in computational time and work achieved through these modifications is illustrated by numerical results.     

Numerical solution of matrix equations of the Stein type in the self-adjoint case

The algorithms for solving the equations X ? AX ^T B = C and X ? AX*B = C proposed by the authors in earlier publications are now modified for the case where these equations can be regarded as self-adjoint ones. The economy in the computational time and work achieved through these modifications is illustrated by numerical results.     

Modifying a numerical algorithm for solving the matrix equation X + AX T B = C

Certain modifications are proposed for a numerical algorithm solving the matrix equation X + AX ^T B = C. By keeping the intermediate results in storage and repeatedly using them, it is possible to reduce the total complexity of the algorithm from O(n ⁴) to O(n ³) arithmetic operations.     

The algebraic Riccati equation: conditions for the existence and uniqueness of solutions

Unmixed solutions of the matrix equation XDX+XA+AX^*?C=0, D?0 are studied.     

线性流形上的广义反射矩阵反问题

设 R∈C^m×m 及 S∈C^n×n 是非平凡Hermitian酉矩阵, 即 R^H=R=R^-1≠±I_m ,S^H=S=S^-1≠±I_n.若矩阵 A∈C^m×n 满足 RAS=A, 则称矩阵 A 为广义反射矩阵.该文考虑线性流形上的广义反射矩阵反问题及相应的最佳逼近问题.给出了反问题解的一般表示, 得到了线性流形上矩阵方程AX₂=Z₂, Y₂^H A=W₂^H 具有广义反射矩阵解的充分必要条件, 导出了最佳逼近问题唯一解的显式表示.     

Roth's similarity theorem and rank minimization in the presence of nonderogatory or semisimple eigenvalues

Roth's similarity theorem on the consistency of Sylvester's matrix equation AX???XA?=?C can be extended to a theorem on rank minimization if the common eigenvalues of A and B are nonderogatory or semisimple.     

The matrix equations AX = C, XB = D

For the pair of matrix equations AX = C, XB = D this paper gives common solutions of minimum possible rank and also other feasible specified ranks.     

On a particular case of the inconsistent linear matrix equation AX+YB=C

We consider the linear matrix equation AX+YB=C where A,B, and C are given matrices of dimensions (r+1)×r, s×(s+1), and (r+1)×(s+1), respectively, and rank A = r, rank B = s. We give a connection between the least-squares solution and the solution which minimizes an arbitrary norm of the residual matrix C?AX? YB.     

The solution of the equation AX + BX ⋆ = 0

We give a complete solution of the matrix equation AX?+?BX ^??=?0, where A, B?∈?? ^m×n are two given matrices, X?∈?? ^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.     

Some extensions of the Kantorovich inequality and statistical applications

Kantorovich gave an upper bound to the product of two quadratic forms, (X′AX) (X′A⁻¹X), where X is an n-vector of unit length and A is a positive definite matrix. Bloomfield, Watson and Knott found the bound for the product of determinants |X′AX| |X′A⁻¹X| where X is n × k matrix such that X′X = I_k. In this paper we determine the bounds for the traces and determinants of matrices of the type X′AYY′A⁻¹X, X′B²X(X′BCX)⁻¹ X′C²X(X′BCX)⁻¹ where X and Y are n × k matrices such that X′X = Y′Y = I_k and A, B, C are given matrices satisfying some conditions. The results are applied to the least squares theory of estimation.     

Existence of a low rank or ℋ︁‐matrix approximant to the solution of a Sylvester equation

We consider the Sylvester equation AX?XB+C=0 where the matrix C∈?^n×m is of low rank and the spectra of A∈?^n×n and B∈?^m×m are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any ε∈(0,1) there exists a matrix X? of rank k=O(log(1/ε)) such that ∥X?X?∥₂?ε∥X∥₂. As a generalization we prove that if A,B,C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format described in Hackbusch (Computing 2000; 62 : 89–108). The blockwise rank of the approximation is again proportional to log(1/ε). Copyright © 2004 John Wiley & Sons, Ltd.     

Conditions for unique solvability of the matrix equation AX + X*B = C

Conditions for the unique solvability of the matrix equation AX + X*B = C are formulated in terms of the eigenvalues and the Kronecker structure of the matrix pencil A + ??B* associated with this equation.     

The matrix equation AX − YB = C

A necessary and sufficient condition is established for solvability of the matrix equation AX ? YB = C. The condition differs from that given by W.E. Roth. The general solution of the equation is also found.