Random Perturbations of Matrix Polynomials

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of the resolvent of the sum is derived and the eigenvalues are localised. Three instances are considered: a low-rank matrix perturbed by the Wigner matrix, a product $HX$ of a fixed diagonal matrix $H$ and the Wigner matrix $X$ and a special matrix polynomial. The results are illustrated with various examples and numerical simulations.Comment: 28 pages, 5 figure

Generic symmetric matrix pencils with bounded rank

We show that the set of n × n complex symmetric matrix pencils of rank at most r is the union of the closures of [r/2] + 1 sets of matrix pencils with some, explicitly described,complete eigenstructures. As a consequence, these are the generic complete eigenstructures of n × n complex symmetric matrix pencils of rank at most r. We also show that these closures correspondto the irreducible components of the set of n × n symmetric matrix pencils with rank at most r when considered as an algebraic set

A new look at pencils of matrix valued functions

AbstractMatrix pencils depending on a parameter and their canonical forms under equivalence are discussed. The study of matrix pencils or generalized eigenvalue problems is often motivated by applications from linear differential-algebraic equations (DAEs). Based on the Weierstrass-Kronecker canonical form of the underlying matrix pencil, one gets existence and uniqueness results for linear constant coefficients DAEs. In order to study the solution behavior of linear DAEs with variable coefficients one has to look at new types of equivalence transformations. This then leads to new canonical forms and new invariances for pencils of matrix valued functions. We give a survey of recent results for square pencils and extend these results to nonsquare pencils. Furthermore we partially extend the results for canonical forms of Hermitian pencils and give new canonicalforms there, too. Based on these results, we obtain new existence and uniqueness theorems for differential-algebraic systems, which generalize the classical results of Weierstrass and Kronecker

A Geometric Description of the Sets of Palindromic and Alternating Matrix Pencils with Bounded Rank

The sets of n x n T-palindromic, T-antipalindromic, T-even, and T-odd matrix pencils with rank at most r < n are algebraic subsets of the set of n x n matrix pencils. In this paper, we determine their dimension and we prove that they are all irreducible. This is in contrast with the nonstructured case, since it is known that the set of n x matrix pencils with rank at most r< n is an algebraic set with r + 1 irreducible components. We also show that these sets of structured pencils with bounded rank are the closure of the congruence orbit of a certain structured pencil given in canonical form. This allows us to determine the generic canonical form of a structured n x n matrix pencil with rank at most r, for any of the previous structures

Miniversal deformations of pairs of symmetric matrices under congruence

For each pair of complex symmetric matrices $(A,B)$ we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices $(\widetilde{A},\widetilde{B})$, close to $(A,B)$ can be reduced by congruence transformation that smoothly depends on the entries of $\widetilde{A}$ and $\widetilde{B}$. Such a normal form is called a miniversal deformation of $(A,B)$ under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from $(A,B)$ to its miniversal deformation.Comment: arXiv admin note: text overlap with arXiv:1104.249