Notation: The set of real symmetric n ×n matrices is denoted S . A matrix A ∈ S is called positive semidefinite if x Ax ≥ 0 for all x ∈ R, and is called positive definite if x Ax > 0 for all nonzero x ∈ R . The set of positive semidefinite matrices is denoted S and the set of positive definite matrices + n is denoted by S++. The cone S is a proper cone (i.e., closed, convex, pointed, and solid). +

In this paper, we show that if an injective map  on symmetric matrices   n S C satisfies then           , , n ABA A B A A B S        C ,   Φ t f A SA S   for all   n A S C  , where f is an injective homomorphism on , is a complex orthogonal matrix and C S f A is the image of A under f applied entrywise.

S. J. Choi and H. Y. Youn proposed a key pre-distribution scheme for Wireless Sensor Networks based on LU decomposition of symmetric matrix, and later many researchers did works based on this scheme. Nevertheless, we find a mathematical relationship of L and U matrixes decomposed from symmetric matrix, by using which we can calculate one matrix from another regardless of their product – the key...

We consider the symmetric Darlington synthesis of a p × p rational symmetric Schur function S with the constraint that the extension is of size 2p×2p. Under the assumption that S is strictly contractive in at least one point of the imaginary axis, we determine the minimal McMillan degree of the extension. In particular, we show that it is generically given by the number of zeros of odd multipli...

The completely 2, symmetric S-matrix defined by Belavin is shown to satisfy the Yang-Baxter equations. In the projective space of Boltzmann weights, the curves on which there exist commuting transfer matrices are shown to be embedded elliptic curves. Explicit polynomial equations for these curves are given. For n = 2 these results reduce to the results of Baxter for the symmetric eight-vertex m...

We consider the problem of finding a point in a nonempty bounded convex body Γ in the cone of symmetric positive semidefinite matrices S + . Assume that Γ is defined by a separating oracle, which, for any given m×m symmetric matrix Ŷ , either confirms that Ŷ ∈ Γ or returns several selected cuts, i.e., a number of symmetric matrices Ai, i = 1, ..., p, p ≤ pmax, such that Γ is in the polyhedron {...

Let Sn(S) denote the set of symmetric matrices over some semiring, S. A line of A ∈ Sn(S) is a row or a column of A. A star of A is the submatrix of A consisting of a row and the corresponding column of A. The term rank of A is the minimum number of lines that contain all the nonzero entries of A. The star cover number is the minimum number of stars that contain all the nonzero entries of A. Th...

We study the eigenvalues and eigenspaces of the quadratic matrix polynomial Mλ + sDλ + K as s → ∞, where M and K are symmetric positive definite and D is symmetric positive semi-definite. The work is motivated by its application to modal analysis of finite element models with strong linear damping. Our results yield a mathematical explanation of why too strong damping may lead to practically un...

The P, Z, and S properties of a linear transformation on a Euclidean Jordan algebra are generalizations of the corresponding properties of a square matrix on R. Motivated by the equivalence of P and S properties for a Z-matrix [2] and a similar result for Lyapunov and Stein transformations on the space of real symmetric matrices [6], [5], in this paper, we present two results supporting the con...