Some Asymptotic Behaviors Associated with Matrix Decomposition

We obtain several asymptotic results on the powers of a square matrix associated with SVD, QR decomposition and Cholesky decomposition. 1. Yamamoto’s theorem Given X ∈ Cn×n, the eigenvalue moduli of X are always less than or equal to the spectral norm of X since ‖X‖ := max ‖v‖2=1 ‖Xv‖2 and if Xv = λv for some unit vector v, then |λ| = ‖Xv‖2 ≤ ‖X‖. So for all m ∈ N, r(X) ≤ ‖X‖ ≤ ‖X‖, where r(X) denotes the spectral radius of X. The following is the celebrated Beruling-Gelfand’s theorem (the finite dimensional case) [9, p.235, p.379]. Theorem 1.1. Let X ∈ Cn×n. then (1.1) lim m→∞ ‖X‖ = r(X). Needless to say, (1.1) is true for all norms since they are all equivalent. We now provide an elementary proof which is different from those in [11, 7, 2, 8]. Proof. Since ‖ · ‖ is invariant under unitary similarity, by Schur triangularization theorem, we may assume that X = T is upper triangular with ascending diagonal moduli |t11| ≤ · · · ≤ |tnn|. When X is nilpotent, that is, r(X) = 0, (1.1) is obviously true. Hence we may assume that X is not nilpotent so that r(X) = |tnn| 6 = 0. Write Tm = [t ij ] ∈ Cn×n which is also upper triangular. For 1 ≤ i ≤ j ≤ n,