On Perturbation bounds for HR decomposition

摘要

In [1], Bunser-Gerstner showed that almost every complex square matrix A cart be decomposed into a product of a so-called pseudo-Hermitian matrix H and an upper triangular matrix:A=HR and gave the necessary conditions for the existence and the uniqueness of HR decomposition. Let J = diag(±1 ) be a signature matrix. Let A be the set of those matrices A for which the leading principal minors of A* J A have the same signs as the corresponding minors of J, R. Bhatia in [2] showed that for any A∈A has a decomposition A= HR,where R is the upper triangular matrices whose diagonal entries are all positive, H satisfies H*JH = J.Let A and A = A + E have the decomposition A = HR and A=HR, R. Bhatia [2] obtained ‖(H)-H‖F≤√(2cond(H)‖R-1 ‖‖(A)-A‖F, (1)‖( R )-R‖F≤(2)cond ( R ) ‖H-1‖‖(A)-A‖F. (2)Where cond(X) =‖1X‖‖X-1‖,‖X‖is the spectral norm, ‖X‖F is the Frobenius norm. Adopting the technique used in [3,4], in this paper we first establish some new first-order bounds of the HR decomposition for A +tG = H(t)R(t), [t]≤ε if ε is small enough that the leading principal minors of (A + tG)T J(A + tG) have the same signs as the corresponding minors of J, and get H(O) = GR-1- H R(O)R-1, (3)R(0) = Jup(R-1 GT JH + Hr jGR-1)R. (4)In particular, when △A = εG, A +△A has the decomposition A +△A = (H + △H)(R + △R) with △H = εH(0) + O(ε2), △R =εk(0) + 0(ε2).Using the row scaling on the perturbation analysis for the sensitivity of R , the more tighter bound than(2) we obtain is‖R(O)‖F/‖R‖F≤KR(A) ‖G‖F/‖H‖F (5)where KR(A)≡inf D∈Dn KR(A,D)≡inf D∈Dn 1+ξ2 K2(D-1 R) ,D =diag(δ1,δ2,…,δn) ~ l) the set of all n×n real positive definitive diagonal matrices and δD - max δj/δi,δi> 0. And ‖H(0)‖F/‖H‖F ≤KH(A)‖G‖F/‖H‖F (6)is also more tighter than (1) for the sensitivity of H. Where K H(A) ≡inf KH(A,D)≡inf 1+δD2 K2(HD-1) ‖R-1‖‖2‖G‖2.The condition estimates for the HR decomposition can also be derived by (5) and (6).