A simple and efficient numerical algorithm for computing the exponential of a symmetric matrix is developed in this paper. For an n× n matrix, the required number of operations is around 10/3 n. It is based on the orthogonal reduction to a tridiagonal form and the Chebyshev uniform approximation of e−x on [0,∞).

The spectral theorem in linear algebra tells us that every symmetric matrix A (n x n) can be factored as A = PDP , where P is an orthogonal matrix and D is a diagonal matrix comprising of the eigenvalues of A. Such a diagonalization is possible only when A is symmetric. What if A is just a square matrix but not symmetric? We know that every square matrix A (symmetric or not) is diagonalizable a...

The contragredient transformation A 7→ PAP, B 7→ PBP of two matrices A,B effects simultaneous similarity transformations of the products AB and BA. This work provides structured canonical forms under this transformation for symmetric or skew-symmetric A,B. As an application, these forms are used to study the quadratic matrix equation XAX = B, where both A,B are skew-symmetric or symmetric matri...

Let P = f" + (I ,,) , the direct sum of the p x p identity matrix and the negative of the q x q ident ity matrix. The fo llowing theorem is proved. TH EOHEM: If X = cZ where Z is a 4 x 4 P-orthogonal , P-skew-symmetric matrix and Ie I .;;; 2, there exist matrices A an.d B, both of which are P-orthogollal and P-skew-symmetric, sach that X = AB BA. Methods for o btaining certain matrices which sa...

The contragredient transformation A 7→ PAP, B 7→ PBP of two matrices A,B effects simultaneous similarity transformations of the products AB and BA. This work provides structured canonical forms under this transformation for symmetric or skew-symmetric A,B. As an application, these forms are used to study the quadratic matrix equation XAX = B, where both A,B are skew-symmetric or symmetric matri...

The contragredient transformation A 7→ P−1AP−T, B 7→ PTBP of two matrices A,B effects simultaneous similarity transformations of the products AB and BA. This work provides structured canonical forms under this transformation for symmetric or skew-symmetric A,B. As an application, these forms are used to study the quadratic matrix equation XAX = B, where both A,B are skew-symmetric or symmetric ...

This paper is concerned with the approximation of the effective conductivity σ(A) associated to an elliptic operator ∇xA(x, η)∇x where for x ∈ R d, d ≥ 1, A(x, η) is a bounded elliptic random symmetric d × d matrix and η takes value in an ergodic probability space. Writing AN (x, η) the periodization of A(x, η) on the torus T d N of dimension d and side N we prove that η-a.s. lim N→+∞ σ(A (x, η...

A set of fixed points of the Hopfield type neural network is under investigation. Its connection matrix is constructed with regard to the Hebb rule from a highly symmetric set of the memorized patterns. Depending on the external parameter the analytic description of the fixed points set has been obtained. A set of fixed points of the Hopfield type neural network is under investigation. Its conn...