The Laurent expansion for a nearly singular matrix

Abstract

Let A0, A1 be n × n matrices of complex numbers and let En be the vector space of n × 1 matrices of complex numbers. Let N1 = \s{x ∈ En|A1x = 0\s}, N0,-1 = \s{0\s} ⊂ En, and for k≥- 1 define R1k = A1N0k and N0k+1 ={x ∈ En{divides}A0x ∈ R1k}. In any case μ = min{k≥ - 1 {divides} N0,k+1 = N0,k exists and μ ≥ 0 or μ = - 1 according as A0 is singular or not. The main result presented is the following: There exists δ> 0 such that the matrix A0 + zA1 is invertible for all complex numbers z such that 0 < |z| < δ if and only if N1 {n-ary intersection} N0k = \s{0\S} for all ≥ 0. Moreover, if this condition holds, then there exist n × n matrices Qk such that, the series converging for 0 < |z|< δ for some δ > 0, and Q-μ-1 ≠ 0. © 1971.