A sharp version of Kahanʹs theorem on clustered eigenvalues Original Research Article

Let the n × n Hermitian matrix A have eigenvalues λ1, λ2, …, λn, let the k × k Hermitian matrix H have eigenvalues μ1 ≤ μ2 … ≤ μk, and let Q be an n × k matrix having full column rank, so 1 ≤ k ≤ n. It is proved that there exist k eigenvalues λi1 ≤ λi2 ≤ … ≤ λik of A such that image always holds with c = 1, where σmin (Q) is the smallest singular value of Q, and double vertical bar; · double vertical bar;2 denotes the biggest singular value of a matrix. The inequality was proved for c ≤ √2 in 1967 by Kahan, who also conjectured that it should be true for c = 1.