Nearest pair with more nonconstant invariant factors and pseudospectrum

Abstract

Let (A,B)∈ C n×n× C n×m . Suppose that the number of nonconstant (i.e., ≠1) invariant factors of the polynomial matrix λ[I n,0]−[A,B] is less than k. For all complex number λ denote by σ n−(k−1)(λ[I n,0]−[A,B]) the greatest (n−(k−1))th singular value of the matrix λ[I n,0]−[A,B] . The minimum absolute value of the real function of complex variable λ↦σ n−(k−1)(λ[I n,0]−[A,B]) gives the distance from (A,B) to the set of pairs with more or equal number of nonconstant invariant factors. When k=1, this specializes in the formula of Eising for the distance from a controllable pair (A,B) to the nearest uncontrollable pair. The complex numbers λ lying in the sublevel set {λ∈ C | σ n (λ[ I n , 0]−[ A, B])⩽ε} of the function λ↦σ n(λ[I n,0]−[A,B]), are the uncontrollable modes of all the pairs that are within an ε tolerance of (A,B). All the results of this paper are an immediate consequence of the Singular Value Decomposition of a matrix and of the interpretation of the singular values as the distances to the nearest matrices of lower ranks.