ON CLOSE EIGENVALUES OF TRIDIAGONAL MATRICES

A symmetric tridiagonal matrix with a multiple eigenvalue must have a zero subdiagonal element beta(i) and must be a direct sum of two compl ementary blocks, both of which have the eigenvalue. Yet it is well kno wn that a small spectral gap does not necessarily imply that some beta (i) is small, as is demonstrated by the Wilkinson matrix. In this note , it is shown that a pair of close eigenvalues can only arise from two complementary blocks on the diagonal, in spite of the fact that the b eta(i) coupling the two blocks may not be small. In particular, some e xplanatory bounds are derived and a connection to the Lanczos algorith m is observed. The nonsymmetric problem is also included.