The Laurent expansion for a nearly singular matrix

AbstractLet 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∣A0x ∈ R1k}. In any case μ = min{k⩾ − 1 ∣ 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 ⋂ 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