For an algebraically closed field F, we show that any matrix polynomialn×m , n ≤ m, can be reduced to triangular form, preserving the degree and the finite and infinite elementary divisors. We also characterize the real matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable with diagonal blocks of sizes 1 × 1 and 2 × 2. The proofs we present solve the structured inverse problem of building up triangular matrix polynomials starting from lists of elementary divisors.

Non-negative matrix factorization (NMF) condenses high-dimensional data into lower-dimensional models subject to the requirement that data can only be added, never subtracted. However, the NMF problem does not have a unique solution, creating a need for additional constraints (regularization constraints) to promote informative solutions. Regularized NMF problems are more complicated than conventional NMF problems, creating a need for computational methods that incorporate the extra constraints in a reliable way. We developed novel methods for regularized NMF based on block-coordinate descent with proximal point modification and a fast optimization procedure over the alpha simplex. Our framework has important advantages in that it (a) accommodates for a wide range of regularization terms, including sparsity-inducing terms like the penalty, (b) guarantees that the solutions satisfy necessary conditions for optimality, ensuring that the results have well-defined numerical meaning, (c) allows the scale of the solution to be controlled exactly, and (d) is computationally efficient. We illustrate the use of our approach on in the context of gene expression microarray data analysis. The improvements described remedy key limitations of previous proposals, strengthen the theoretical basis of regularized NMF, and facilitate the use of regularized NMF in applications.

Abstract. We consider quadratic eigenproblems Mλ 2 + Dλ + K x = 0, where all coefficient matrices are real and positive semidefinite, (M, K) is regular, and D is of low rank. Matrix polynomials of this form appear in the analysis of vibrating structures with discrete dampers. We develop an algorithm for such problems, which first solves the undamped problem Mλ 2 + K x = 0 and then accommodates for the low rank term Dλ. For the first part, we develop a new algorithm based on a method proposed by Wang and Zhao [SIAM J. Matrix Anal. Appl., 12 (1991), pp. 654-660], which can compute all eigenvalues of definite generalized eigenvalue problems with semidefinite coefficient matrices in a backward stable and symmetry preserving manner. We use this new algorithm to compute the solution to the undamped problem, and then we use this solution in order to compute all eigenvalues of the original problem and the associated eigenvectors if requested. To this end, we use an Ehrlich-Aberth iteration that works exclusively with vectors and tall skinny matrices and contributes only lower order terms to the overall flop count. Numerical experiments show that the proposed algorithm is both fast and accurate. Finally we discuss the application to large scale quadratics and the possibility of generalizations to other problems.

Abstract. We study the eigenvalues and eigenspaces of the quadratic matrix polynomial Mλ 2 + sDλ + K as s → ∞, where M and K are symmetric positive definite and D is symmetric positive semidefinite. This work is motivated by its application to modal analysis of finite element models with strong linear damping. Our results yield a mathematical explanation of why too strong damping may lead to practically undamped modes such that all nodes in the model vibrate essentially in phase.Key words. quadratic eigenvalue problem, principal angles, canonical angles, matrix polynomial, viscous damping, discrete damper, vibrating system AMS subject classifications. 15A18, 15A22, 65F15, 70J10, 70J30, 70J50 DOI. 10.1137/140959390 1. Introduction. A way to prevent a structure from vibrating violently is to incorporate viscous dampers into the design. A viscous damper is a device that resists motion by producing a force proportional to the velocity of a piston relative to its housing (see Figure 1) raised to a power α. In this paper we consider linear damping, which corresponds to dampers with α = 1. This value of α is the default for certain product lines of seismic dampers [1]. The resisting force produced by a viscous damper arises when fluid, trapped in a cylinder, is forced through small holes.By adjusting the size of these holes, we can make the damper stronger. But stronger is not necessarily better: if a damper is too strong, it resembles a rigid component and hence has little purpose. This suggests that a structure with only very strong dampers should be quite similar to a structure without dampers. The goal of this paper is to investigate this phenomenon more rigorously for discretized structures. We will do this by studying the eigenvalues and eigenspaces of a related quadratic matrix polynomial.Consider a finite element model of a structure with r viscous dampers. If the model vibrates freely, the displacements of its nodes are given by the solutions to the equations of motion:

We perform a backward error analysis of the inexact shift-and-invert Arnoldi algorithm. We consider inexactness in the solution of the arising linear systems, as well as in the orthonormalization steps, and take the non-orthonormality of the computed Krylov basis into account. We show that the computed basis and Hessenberg matrix satisfy an exact shift-and-invert Krylov relation for a perturbed matrix, and we give bounds for the perturbation. We show that the shift-and-invert Arnoldi algorithm is backward stable if the condition number of the small Hessenberg matrix is not too large. This condition is then relaxed using implicit restarts. Moreover, we give notes on the Hermitian case, considering Hermitian backward errors, and finally, we use our analysis to derive a sensible breakdown condition.