Computing lower rank approximations of matrix polynomials (Q2284958)

The contribution deals with the approximation of matrix polynomials with floating point coefficients by means of other matrix polynomials of a given maximal rank. A matrix polynomial is a single-variable polynomial whose coefficients are matrices (in this case with real number entries). Given such a matrix polynomial \(P\) of rank \(n\) and an integer \(k \leq n\), the task is to determine a minimal (according to a specific metric) matrix polynomial \(\Delta\) such that \(P-\Delta\) is a matrix polynomial of rank \(n - k\). The specific metric discussed here is derived from the Frobenius norm. Overall, the paper deals with the geometry of those minimal solutions. It first establishes that the task is well-posed (i.e., such minimal matrix polynomials \(\Delta\) exist) and that the solutions are isolated (i.e., surrounded by a nontrivial open neighbourhood of non-minimal solutions). Several additional properties are also discussed. An iterative procedure to efficiently obtain such a minimal lower-rank approximation is also presented and was implemented in Maple. Convergence and numerical robustness are showcased on the implementation. The paper is well written, but assumes a decent exposure to linear algebra, optimization theory and topology. Proofs' details are provided and some final examples illustrate the algorithm. With some effort, a suitably prepared reader can certainly appreciate this contribution.