Christof Vermeersch (KU Leuven): Current and Future Perspectives on the Block Macaulay Matrix

Multiparameter eigenvalue problems generalize one-parameter eigenvalue problems by introducing multiple spectral parameters. While these problems may be unfamiliar to the broader scientific community, they are well-known territory here at FMF in Ljubljana. These problems arise naturally in various application areas, for example, when identifying globally optimal model parameters from input-output data or higher-order transfer functions. The block Macaulay matrix—a sparse and structured matrix constructed from the problem's coefficient matrices—can be exploited to compute the eigenvalues of a multiparameter eigenvalue problem and, for example, retrieve the unknown model parameters.

In this talk, I first introduce the different subspaces of the block Macaulay matrix and explain how their multi-shift-invariant structure gives rise to a joint generalized eigenvalue problem, the eigenvalues of which correspond to the multiparameter eigenvalue solutions. Current block Macaulay algorithms are inspired by resultant-based polynomial root-finding methods and combine techniques from systems theory and numerical linear algebra. Next, I explore several promising future research directions: combining subspaces from multiple block Macaulay matrices, finding connections with other solution algorithms, and leveraging the inherent bihomogeneous problem structure. These advances could help to tackle more complex applications in the (near) future!

This talk presents joint efforts with and insights from Bart De Moor, Bor Plestenjak, and Matías Bender.