Nearest pair with more nonconstant invariant factors and pseudospectrum Original Research Article

Let image . Suppose that the number of nonconstant (i.e., ≠1) invariant factors of the polynomial matrix image λ[In,0]−[A,B]is less than k. For all complex number λ denote by image σn−(k−1)(λ[In,0]−[A,B])the greatest (n−(k−1))th singular value of the matrix λ[In,0]−[A,B]. The minimum absolute value of the real function of complex variable image λmaps toσn−(k−1)(λ[In,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 image of the function image λmaps toσn(λ[In,0]−[A,B]),are the uncontrollable modes of all the pairs that are within an var epsilon 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.