稳定的不变子空间的扰动界限

PERTURBATION BOUNDS FOR STABLE INVARIANT SUBSPACES

§1.引言 1.1.稳定的不变子空间 在矩阵的各类不变子空间中,从扰动分析的角度研究得比较深入的,是由根子空间的直和构成的不变子空间。

Let A be an n×n complex matrix having different eigenvalues λ_1,…,λ_r, andlet dim N(A—λ_1I)=1 for i=1,…,s (≤r) and dim N(A—λ_iI)>1 for i=s+1,…,r. Let x=x?x be an l-dimensional stable invariant subspaceof A, wkere x=x_1?…?x_(t1) (s_1≤s) in which for each i(1≤i≤s_1) thesubspace x_i is an arbitrary A-invariant subspace of N((A—λ,I)'), and X(2)is the direct sum of the root subspaces for A corresponding to some of the eigenva-1ues 1,.+l,'..,1.. This paper gives perturbation bounds of the subspace X when Ais perturbed. As a consequence it is shown that there exist positive constants c and8 such that every n X n complex matrix A with ljA -- AlI wt B has an l- dimensio-nal invariant subspace X satisfyingl1JPE -- Pxll < clf Z -- AlM,where ll. ll is an arbitrary unitarily invariant' norm,and m(1) is the largest algebraicmu1tiplicity of 1l,''',1,,. Besides, a lower bound of the distance betweeR a nondero-gatory matrix and the set of derogatory matrices is also given.