EXTENDED INTERLACING INTERVALS

Classical interlacing for a Hermitian matrix A may be viewed as descri bing how many eigenvalues of A must be captured by intervals determine d by eigenvalues of a principal submatrix (A) over cap of A. If the si ze (A) over cap is small relative to that of A, then it may be that no eigenvalues of A are guaranteed to be in an interval determined by on ly a few consecutive eigenvalues of (A) over cap. Here, we generalize classical interlacing theorems by using singular values of off-diagona l blocks of A to construct extended intervals that capture a larger nu mber of eigenvalues of A. In the event that an appropriate off-diagona l block has low rank, the extended interval may be no wider, giving st ronger statements than classical interlacing. The union of pairs of in tervals is also discussed, and some applications of the ideas are ment ioned. (C) 1997 Elsevier Science Inc., 1997.