Two canonical forms for skew-symmetric matrix polynomials over arbitrary fields are characterized — the Smith form, and its skew-symmetric variant obtained via unimodular congruences. Applications include the analysis of the eigenvalue and elementary divisor structure of products of two skew-symmetric matrices, the derivation of a Smith-McMillan-like canonical form for skew-symmetric rational m...

We present a framework for accelerating a spectrum of machine learning algorithms that require computation of bilinear inverse forms u>A u, where A is a positive definite matrix and u a given vector. Our framework is built on Gausstype quadrature and easily scales to large, sparse matrices. Further, it allows retrospective computation of lower and upper bounds on u>A u, which in turn accelerate...

This paper presents equivalent forms of the Sylvester matrix equations. These equivalent forms allow us to use block linear methods for solving large Sylvester matrix equations. In each step of theses iterative methods we use global FOM or global GMRES algorithm for solving an auxiliary block matrix equations. Also, some numerical experiments for obtaining the numerical approximated solution of...

For a class of skew-normal matrix distributions, the density function, moment generating function and independent conditions are obtained. The noncentral skew Wishart distribution is defined and the necessary and sufficient conditions under which a quadratic form is noncentral skew Wishart distributed random matrix are established. A new version of Cochran’s theorem is given. For illustration, ...

We propose a computationally efficient and numerically reliable algorithm to compute Kronecker-like forms of periodic matrix pairs. The eigenvalues and Kronecker indices are defined via the Kro-necker structure of an associated lifted matrix pencil. The proposed reduction method relies on structure preserving manipulations of this pencil to extract successively lower complexity subpencils which...

We give modular algorithms to compute row-reduced forms, weak Popov forms, and Popov forms of polynomial matrices, as well as the corresponding unimodular transformation matrices. Our algorithms improve on existing fraction-free algorithms. In each case we define lucky homomorphisms, determine the appropriate normalization, as well as bound the number of homomorphic images required. The algorit...

The standard way to solve polynomial eigenvalue problems P (λ)x = 0 is to convert the matrix polynomial P (λ) into a matrix pencil that preserves its spectral information– a process known as linearization. When P (λ) is palindromic, the eigenvalues, elementary divisors, and minimal indices of P (λ) have certain symmetries that can be lost when using the classical first and second Frobenius comp...

In a design process products can be described by the requests of users, the requirements of design, functional structures, sketches and technical drawings [1]. The complexity, one of the main problems of conceptual design [2], is included into the modelling and representation of a design solution. To represent a design solution in its conceptual phase, and are of particular importance. The mean...

A generalized matrix product can be formally written as A sp p A sp−1 p−1 · · ·A s2 2 A s1 1 , where si ∈ {−1,+1} and (A1, . . . , Ap) is a tuple of (possibly rectangular) matrices of suitable dimensions. The periodic eigenvalue problem related to such a product represents a nontrivial extension of generalized eigenvalue and singular value problems. While the classification of generalized matri...

We present a variant of the Lagrange-Gauss reduction of quadratic forms designed to minimize the norm of the reduction matrix within a quadratic complexity. The matrix computed by our algorithm on the input f has norm O f 1 2 Δ 4 f , which is the square root of the best previously known bounds using classical algorithms. This new bound allows us to fully prove the heuristic lattice based attack...