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 ...

We propose a two-level iterative scheme for solving general sparse linear systems. The proposed consists of preconditioner that increases the norm skew-symmetric part relative to rest and makes main diagonal coefficient matrix as close identity possible so preconditioned system is shifted possible. then solved via particular Minimal Residual Method Shifted Skew-Symmetric Systems (MRS). This lea...

Composite linear and quadratic systems produce three-wave coupling when stimulated by random-phase input sinusoids. Due to the non-linearity of the system the output frequencies are arithmetically related to the input. Using third order cumulant statistics and their associated bispectrum, techniques are devised based on phaseinsensitive matrix structures for detection and frequency estimation o...

The time derivative of a rotation matrix equals the product of a skew-symmetric matrix and the rotation matrix itself. This article gives a brief tutorial on the well-known result.

The covariance matrix of a pattern is composed by its second order central moments. For a rotationally symmetric shape, its covariance matrix is a scalar identity matrix. In this work, we apply this property to restore the skewed shape of rotational symmetry. The relations between the skew transformation matrix and the covariance matrices of original and skewed shapes are derived. By computing ...

In the space ?d, d ? 2, we study diffusion equation ?div (?u? +b(x/?)?u?)+u? = f ?L2(?d), u?H1(?d), where b(y) is an unbounded 1-periodic skew symmetric matrix and ? a small parameter. The assumed to be integrable with respect period exponent s, s for 3 > 2 2. Assuming that solution not necessarily unique, find asymptotics so-called approximate resolvent remainder of order ?2 as ? 0.

While there is no lack of efficient Krylov subspace solvers for Hermitian systems, few exist for complex symmetric, skew symmetric, or skew Hermitian systems, which are increasingly important in modern applications including quantum dynamics, electromagnetics, and power systems. For a large, consistent, complex symmetric system, one may apply a non-Hermitian Krylov subspace method disregarding ...

We introduce a function on sequences, which we call the Pfaffian transform, using the Pfaffian of a skew-symmetric matrix. We establish several basic properties of the Pfaffian transform, and we use the transfer matrix method to show that the set of sequences with rational generating functions is closed under the Pfaffian transform. We conclude by computing the Pfaffian transform of a variety o...

In this paper we consider the solution of linear systems of saddle point type by preconditioned Krylov subspace methods. A preconditioning strategy based on the symmetric/ skew-symmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matrix are established. The potential of this approach is illustrated by numerical experiments with matrices fro...