Homotopy Method for General $\lambda $-Matrix Problems

This paper describes a homotopy method used to solve the kth-degree $\lambda $-matrix problem $( A_k \lambda ^k + A_{k - 1} \lambda ^{k - 1} + \cdots + A_1 \lambda + A_0 )x = 0$. A special homotopy equation is constructed for the case where all coefficients are general $n\times n$ complex matrices. Smooth curves connecting trivial solutions to desired eigenpairs are shown to exist. The homotopy equations maintain the nonzero structure of the underlying matrices (if there is any) and the curves correspond only to different initial values of the same ordinary differential equation. Therefore, the method might be used to find all isolated eigenpairs for large-scale $\lambda $-matrix problems on single-instruction multiple data (SIMD) machines.