Residual bounds for unitarily invariant norms on clustered eigenvalues Original Research Article

Let n × n Hermitian matrix A have eigenvalues λ1, λ2, …, λn, let 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 = 2, where σmin(Q) is the smallest singular value of Q and · denotes any unitarily invariant norm. The assumptions Q*Q = I and H = Q*AQ in Stewart and Sunʹs book are deleted. We improve it applicable to practical computation.