On Wilkinson's problem for matrix pencils

Suppose that an n-by-n regular matrix pencil A − λB has n distinct eigenvalues. Then determining a defective pencil E−λF which is nearest to A−λB is widely known as Wilkinson’s problem. It is shown that the pencil E − λF can be constructed from eigenvalues and eigenvectors of A − λB when A − λB is unitarily equivalent to a diagonal pencil. Further, in such a case, it is proved that the distance from A − λB to E − λF is the minimum “gap” between the eigenvalues of A − λB. As a consequence, lower and upper bounds for the “Wilkinson distance” d(L) from a regular pencil L(λ) 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(λ).