Let A be a skew matrix of order n over an ordered field. There Is a finite class of skew matrices A such that XA = Y and XA = Y have the same solution sets, where x = y. and y = x for some Indices 1 (perhaps none) and x = x , y = y for the remaining 1 . We show that for each Index h, 1 < h < n, there exists an A such that ä > 0 for all J . A proof of von Neumann's Mlnlmax Theorem for symmetric ...

We consider the direct solution of sparse skew symmetric matrices. We see that the pivoting strategies are similar, but simpler, to those used in the factorization of sparse symmetric indefinite matrices, and we briefly describe the algorithms used in a forthcoming direct code based on multifrontal techniques for the factorization of real skew symmetric matrices. We show how this factorization ...

Efficient, backward-stable, doubly structure-preserving algorithms for the Hamiltonian symmetric and skew-symmetric eigenvalue problems are developed. Numerical experiments confirm the theoretical properties of the algorithms. Also developed are doubly structure-preserving Lanczos processes for Hamiltonian symmetric and skew-symmetric matrices.

Let $S(G^{sigma})$ be the skew-adjacency matrix of the oriented graph $G^{sigma}$, which is obtained from a simple undirected graph $G$ by assigning an orientation $sigma$ to each of its edges. The skew energy of an oriented graph $G^{sigma}$ is defined as the sum of absolute values of all eigenvalues of $S(G^{sigma})$. Two oriented graphs are said to be skew equienergetic iftheir skew energies...

An integral square matrix A is called principally unimodular (PU if every nonsingular principal submatrix is unimodular (that is, has determinant \1). Principal unimodularity was originally studied with regard to skew-symmetric matrices; see [2, 4, 5]; here we consider symmetric matrices. Our main theorem is a generalization of Tutte's excluded minor characterization of totally unimodular matri...

Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we generalise these results to the skew-symmetrizable case by introducing a marking on the exchange graph. Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how...

0 = d dt ‖x(t)‖ = 2x(t) ẋ(t) = 2x(t)Ax(t) = x(t) (A+ A )x(t) for all x(t), which occurs if and only A+A = 0, which is the same as A = −A, i.e., A is skew-symmetric. There are many other ways to see this. For example, the norm of the state will be constant provided the velocity vector is always orthogonal to the position vector, i.e., ẋ(t)x(t) = 0. This also leads us to A + A = 0. Another approa...

We determine a term order on the monomials in the variables Xij , 1 ≤ i < j ≤ n, such that corresponding initial ideal of the ideal of Pfaffians of degree r of a generic n by n skew-symmetric matrix is the Stanley-Reisner ideal of a join of a simplicial sphere and a simplex. Moreover, we demonstrate that the Pfaffians of the 2r by 2r skew-symmetric submatrices form a Gröbner basis for the given...

Efficient, backward-stable, doubly structure-preserving algorithms for the Hamiltonian symmetric and skew-symmetric eigenvalue problems are developed. Numerical experiments confirm the theoretical properties of the algorithms. Also developed are doubly structure-preserving Lanczos processes for Hamiltonian symmetric and skew-symmetric matrices.