On Wilkinson's problem for matrix pencils

Abstract

Suppose that an n-by-n regular matrix pencil A -\lambda B has n distinct eigenvalues. Then determining a defective pencil E−\lambda F which is nearest to A−\lambda B is widely known as Wilkinson’s problem. It is shown that the pencil E −\lambda F can be constructed from eigenvalues and eigenvectors of A −\lambda B when A − \lambda B is unitarily equivalent to a diagonal pencil. Further, in such a case, it is proved that the distance from A −\lambda B to E − \lambdaF is the minimum “gap” between the eigenvalues of A − \lambdaB. As a consequence, lower and upper bounds for the “Wilkinson distance” d(L) from a regular pencil L(\lambda) with distinct eigenvalues to the nearest non-diagonalizable pencil are derived.Furthermore, it is shown that d(L) is almost inversely proportional to the condition number of the most ill-conditioned eigenvalue of L(\lambda).