Two complementary block Macaulay matrix algorithms to solve multiparameter eigenvalue problems
Christof Vermeersch, Bart De Moor · Linear Algebra and its Applications · 2022
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.