We give a polynomial-time dynamic programming algorithm for solving the linear complementarity problem with tridiagonal or, more generally, Hessenberg P-matrices. We briefly review three known tractable matrix classes and show that none of them contains all tridiagonal P-matrices.

We show that for certain systems of linear equations with coefficient matrices of Hessenberg form it is possible to use Gaussian elimination to obtain an extrapolated version of the Gauss-Seidel iterative process where the iteration matrix has spectral radius zero. Computational aspects of the procedure are discussed.

——————— Abstract: We investigate the zero–patterns that can be created by unitary similarity in a given matrix, and the zero–patterns that can be created by simultaneous unitary similarity in a given sequence of matrices. The latter framework allows a “simultaneous Hessenberg” formulation of Pati’s tridiagonal result for 4×4 matrices. This formulation appears to be a strengthening of Pati’s the...

Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety which arise in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all classical types. This paving is in fact the intersection of a particular Bruhat decomposition with the Hes...

Abstract. We study subvarieties of the flag variety defined by certain linear conditions. These subvarieties are called Hessenberg varieties and arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenberg varieties over GLn(C) and show that they have no odd-dimensional homology. We provide an...

In this paper we give a survey of some results concerning the computation of quadrature formulas on the unit circle. Like nodes and weights of Gauss quadrature formulas (for the estimation of integrals with respect to measures on the real line) can be computed from the eigenvalue decomposition of the Jacobi matrix, Szegő quadrature formulas (for the approximation of integrals with respect to me...

In this paper we show how to perform the Hessenberg reduction of a rank structured matrix under unitary similarity operations in an efficient way, using the Givens-weight representation which we introduced in an earlier paper. This reduction can be used as a first step for eigenvalue computation. We also show how the algorithm can be modified to compute the bidiagonal reduction of a rank struct...

We characterize matrices A ∈ Cn×n whose zero/nonzero pattern requires that the controllability matrix [b Ab Ab . . . An−1b] ∈ Cn×n is of full rank, where b ∈ Cn×1 has exactly one nonzero entry. When all the diagonal entries of A are nonzero we show that this occurs if and only if QAQ is unreduced upper Hessenberg, with Q being a permutation matrix for which Qb = [b1, 0, . . . , 0] T . We also c...

Eigenvalue computations for structured rank matrices are the subject of many investigations nowadays. There exist methods for transforming matrices into structured rank form, QR-algorithms for semiseparable and semiseparable plus diagonal form, methods for reducing structured rank matrices efficiently to Hessenberg form and so forth. Eigenvalue computations for the symmetric case, involving sem...

The implicit Q-theorem for Hessenberg matrices is a widespread and powerful theorem. It is used in the development of for example implicit QR-algorithms to compute the eigendecomposition of Hessenberg matrices. Moreover it can also be used to prove the essential uniqueness of orthogonal similarity transformations of matrices to Hessenberg form. The theorem is also valid for symmetric tridiagona...