Extended interlacing intervals, Original Research Article

Classical interlacing for a Hermitian matrix A may be viewed as describing how many eigenvalues of A must be captured by intervals determined by eigenvalues of a principal submatrix  of A. If the size  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 only a few consecutive eigenvalues of Â. Here, we generalize classical interlacing theorems by using singular values of off-diagonal blocks of A to construct extended intervals that capture a larger number of eigenvalues of A. In the event that an appropriate off-diagonal block has low rank, the extended interval may be no wider, giving stronger statements than classical interlacing. The union of pairs of intervals is also discussed, and some applications of the ideas are mentioned.