We investigate the matrix equation X - AXB = C. For convenience, the matrix equation X - AXB = C is named as Kalman-Yakubovich-conjugate matrix equation. The explicit solution is constructed when the above matrix equation has unique solution. And this solution is stated as a polynomial of coefficient matrices of the matrix equation. Moreover, the explicit solution is also expressed by the symme...

Suppose that the matrix equation AXB = C with unknown matrix X is given, where A, B, and C are known matrices of suitable sizes. The matrix nearness problem is considered over the general and least squares solutions of the matrix equation AXB = C when the equation is consistent and inconsistent, respectively. The implicit form of the best approximate solutions of the problems over the set of sy...

In this paper, a new method based on parametric form for approximate solu-tion of fuzzy linear matrix equations (FLMEs) of the form AX = B; where Ais a crisp matrix, B is a fuzzy number matrix and the unknown matrix X one,is presented. Then a numerical example is presented to illustrate the proposedmodel.

The vacuum solution ds = dx + x dy + 2 dz dt + lnx dt of the Einstein gravitational field equation follows from the general ansatz ds = dx + gαβ(x) dx dx but fails to follow from it if the symmetric matrix gαβ(x) is assumed to be in diagonal form. KEY: Vacuum solution, Einstein field equation, symmetries, diagonalization

STRUCTURED CONDITIONING OF MATRIX FUNCTIONS∗ PHILIP I. DAVIES† Abstract. The existing theory of conditioning for matrix functions f(X):Cn×n → Cn×n does not cater for structure in the matrix X. An extension of this theory is presented in which when X has structure, all perturbations of X are required to have the same structure. Two classes of structured matrices are considered, those comprising ...

A matrix A = (aij) ∈ Rn×n is said to be symmetric and antipersymmetric matrix if aij = aji = −an−j+1,n−i+1 for all 1 ≤ i, j ≤ n. Peng gave the bisymmetric solutions of the matrix equation A1X1B1+A2X2B2+. . .+AlXlBl = C, where [X1, X2, . . . , Xl] is a real matrices group. Based on this work, an adjusted iterative method is proposed to find the symmetric and antipersymmetric solutions of the abo...

The (matricial) solution set of a Linear Matrix Inequality (LMI) is a convex non-commutative basic open semi-algebraic set (defined below). The main theorem of this paper is a converse, a result which has implications for both semidefinite programming and systems engineering. A non-commutative basic open semi-algebraic set is defined in terms of a non-commutative `×`-matrix polynomial p(x1 · · ...

it is proved that by using bounds of eigenvalues of an interval matrix, someconditions for checking positive deniteness and stability of interval matricescan be presented. these conditions have been proved previously with variousmethods and now we provide some new proofs for them with a unity method.furthermore we introduce a new necessary and sucient condition for checkingstability of interv...

This paper focuses on the study of the second-order directional derivative of a symmetric matrix-valued function of the form F (X) = Pdiag[f(λ1(X)), · · · , f(λn(X))]P . For this purpose, we first adopt a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a matrix in Torki [13]; Second, we establish a formula for the (parabolic) second-order direct...

We present a method for computing the eigenelements of a symmetric matrix A. This method consists in expressing A in the form A = QXQ T , where Q is an orthonormal matrix and X has nonzero components only on main and cross diagonals. The convergence analysis, a comparison with the subspace method and a numerical experiments on a parallel machine are set out.