For a given real invertible skew-symmetric matrix H, we characterize the real 2n×2n matrices X that allow an H-Hamiltonian polar decomposition of the type X = UA, where U is a real H-symplectic matrix (UTHU = H) and A is a real H-Hamiltonian matrix (HA = −ATH).

The solution from the previous section can then be used by defining X to be the matrix in RU×U such that Xw = w ? x. It follows that the kernel matrix K belongs to RU×U and the dual variable α is a signal in RU . The key to the correlation filter is that the circulant structure of X enables the solution to be computed efficiently in the Fourier domain. The matrix X has elements X[u, t] = x[u+ t...

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...

We study how small perturbations of a skew-symmetric matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of skewsymmetric matrix pencil orbits and bundles. In other words, we investigate when the closure of the congruence orbit (or bundle) of a skew-symmetric matrix pencil contains the congruence orbit (or bundle) of another ske...

A spectral function of a Hermitian matrix X is a function which depends only on the eigenvalues of X, 1 (X) 2 (X) : : : n (X), and hence may be written f(1 (X); 2 (X); : : :; n (X)) for some symmetric function f. Such functions appear in a wide variety of matrix optimization problems. We give a simple proof that this spectral function is diierentiable at X if and only if the function f is diier...

For a symmetric positive semidefinite linear system of equations Qx = b, where x = (x1, . . . , xs) is partitioned into s blocks, with s ≥ 2, we show that each cycle of the classical block symmetric Gauss-Seidel (block sGS) method exactly solves the associated quadratic programming (QP) problem but added with an extra proximal term of the form 12‖x−x ‖T , where T is a symmetric positive semidef...

In this paper we study nonlinear problems for Ornstein-Uhlenbeck operators A△v(x) + 〈Sx,∇v(x)〉+ f(v(x)) = 0, x ∈ R, d > 2, where the matrix A ∈ R is diagonalizable and has eigenvalues with positive real part, the map f : R → R is sufficiently smooth and the matrix S ∈ R in the unbounded drift term is skew-symmetric. Nonlinear problems of this form appear as stationary equations for rotating wav...

where N = {1, . . . , n}. In this paper a pattern is assumed to include all diagonal positions. A symmetric pattern is a pattern with the property that (i, j) is in the pattern if and only if (j, i) is also in the pattern; symmetric patterns are also called positionally or combinatorially symmetric. An asymmetric pattern is a pattern with the property that if (i, j) is in the pattern, then (j, ...

Matrix transpose in parallel systems typically involves costly all-to-all communications. In this paper, we provide a comparative characterization of various efficient algorithms for transposing small and large matrices using the popular symmetric multiprocessors (SMP) architecture, which carries a relatively low communication cost due to its large aggregate bandwidth and lowlatency inter-proce...

Consider a convex set S = {x ∈ D : G(x) o 0} where G(x) is a symmetric matrix whose every entry is a polynomial or rational function, D ⊆ R is a domain on which G(x) is defined, and G(x) o 0 means G(x) is positive semidefinite. The set S is called semidefinite representable if it equals the projection of a higher dimensional set which is defined by a linear matrix inequality (LMI). This paper s...