Abstract The partition function of the Symmetric Matrix Ensemble is identified with $$\tau $$ ? -function a particular solution Pfaff Lattice. We show that, in case even power interactions, thermodynamic limit, corresponds to an integrable chain hydrodynamic type. prove that so obtained diagonalisable and adm...

For a given n by n matrix the ratio between the componentwise distance to the nearest singular matrix and the inverse of the optimal Bauer-Skeel condition number cannot be larger than (3 + 2 √ 2) · n. In this note a symmetric matrix is presented where the described ratio is equal to n for the choice of most interest in numerical computation, for relative perturbations of the individual matrix c...

We study the asymmetric respectively symmetric conflict matrix of a multi-clause-set F , where the entry at position (i, j) is the number of literals, which appear positively in clause Ci of F and negatively in clause Cj (at the same time), respectively the number of clashes (at all) between Ci and Cj . A central problem is the determination of the symmetric/asymmetric conflict number of a (sym...

In this paper, characterizations of secondary κkernel symmetric fuzzy matrices are obtained. Relation between sκkernel symmetric, skernel symmetric, κkernel symmetric and kernel symmetric matrices are discussed. Necessary and sufficient conditions are determined for a matrix to be sκkernel symmetric.

In this paper, we investigate the existence of a positive solution of fully fuzzy linear equation systems where fuzzy coefficient matrix is a positive matrix. This paper mainly discusses a new decomposition of a nonsingular fuzzy matrix, a symmetric matrix times to a triangular (ST) decomposition. By this decomposition, every nonsingular fuzzy matrix can be represented as a product of a fuzzy s...

This paper shows that Cayley Transforms, which map Orthogonal and SkewSymmetric matrices, may be considered the extension to matrix field of the complex conformal mapping function f1(z) = 1− z 1 + x . Then, by using a set of real matrices which are, simultaneously, Orthogonal and Symmetric (the Ortho−Sym matrices), it similarly shows how to extend two complex conformal mapping functions (namely...

Given an undamped gyroscopic system GðλÞ 1⁄4 Mλ þ CλþK with M , K symmetric and C skew-symmetric, this paper presents a real-valued spectral decomposition of GðλÞ by a real standard pair ðX;TÞ and a skew-symmetric parameter matrix S . When T is assumed to be a block diagonal matrix, the parameter matrix S has a special structure. This spectral decomposition is applied to solve the quadratic inv...

In this paper, the relationship between the (P,Q)-orthogonal symmetric and symmetric matrices is derived. By applying the generalized singular value decomposition, the general expression of the least square (P,Q)-orthogonal symmetric solutions for the matrix equation AXB = C is provided. Based on the projection theorem in inner space, and by using the canonical correlation decomposition, an ana...

The set of all solutions to the homogeneous system of matrix equations (XA + AX,XB +BX) = (0,0), where (A,B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A,B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. K̊agström, and V.V. Sergeichuk. Skew-symmetric matrix pencils: Codi...

In this work, we investigate several natural computational problems related to identifying symmetric signings of symmetric matrices with specific spectral properties. We show NP-completeness for verifying whether an arbitrary matrix has a symmetric signing that is positive semi-definite, is singular, or has bounded eigenvalues. We exhibit a stark contrast between invertibility and the above-men...