Two complementary block Macaulay matrix algorithms to solve multiparameter eigenvalue problems

Abstract

We consider two algorithms that use the block Macaulay matrix to solve (rectangular) multiparameter eigenvalue problems (MEPs). On the one hand, a multidimensional realization problem in the null space of the block Macaulay matrix constructed from the coefficient matrices of an MEP results in a standard eigenvalue problem (SEP), the eigenvalues and eigenvectors of which yield the solutions of that MEP. On the other hand, we propose a complementary algorithm to solve MEPs that considers the data in the column space of the sparse and structured block Macaulay matrix directly, avoiding the computation of a numerical basis matrix of the null space. This paper generalizes, in a certain sense, traditional Macaulay matrix techniques from multivariate polynomial system solving to the block Macaulay matrix in the MEP setting.