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 Eâ\lambda F which is nearest to Aâ\lambda B is widely known as Wilkinsonâs problem. It is shown that the pencil E â\lambda F can be constructed from eigenvalues and eigenvectors of A â\lambda B when A â \lambda B is unitarily equivalent to a diagonal pencil. Further, in such a case, it is proved that the distance from A â\lambda B to E â \lambdaF is the minimum âgapâ between the eigenvalues of A â \lambdaB. As a consequence, lower and upper bounds for the âWilkinson distanceâ 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).