Interlacing inequalities and control theory

Generalized Interlacing Inequalities

We discuss some applications of generalized interlacing inequalities of Ky Fan to the study of (a) some classical matrix inequalities and (b) matrix problems in quantum information science. AMS Classification 15A18, 15A57, 15A60, 15A90.

Interlacing Inequalities for Totally Nonnegative Matrices

Suppose λ1 ≥ · · · ≥ λn ≥ 0 are the eigenvalues of an n × n totally nonnegative matrix, and λ̃1 ≥ · · · ≥ λ̃k are the eigenvalues of a k × k principal submatrix. A short proof is given of the interlacing inequalities: λi ≥ λ̃i ≥ λi+n−k, i = 1, . . . , k. It is shown that if k = 1, 2, n− 2, n− 1, λi and λ̃j are nonnegative numbers satisfying the above inequalities, then there exists a totally nonneg...

Polynomials , roots , and interlacing

Interlacing Eigenvalues and Graphs

We give several old and some new applications of eigenvalue interlacing to matrices associated to graphs. Bounds are obtained for characteristic numbers of graphs, such as the size of a maximal (colclique, the chromatic number, the diameter, and the bandwidth, in terms of the eigenvalues of the Standard adjacency matrix or the Laplacian matrix. We also deal with inequalities and regularity resu...

Separation and Interlacing Theorems