Block-matrix pseudoinverse characterization

Determine a general formula for computing the Moore–Penrose pseudoinverse of a block matrix in terms of its original matrix blocks, thereby advancing an area that remains technically involved and unresolved.

Background

The convergence analysis of Petrov–Galerkin operator inference requires understanding the pseudoinverse of a block data matrix formed from projected state and input data. The authors note that expressing this pseudoinverse directly through the constituent blocks is difficult. This issue is relevant to deriving general error bounds beyond the special scalar-input Galerkin setting treated in the paper.

References

In general, computing the pseudoinverse of a block matrix in terms of the original matrix blocks is involved and still an active research area.

— Petrov-Galerkin operator inference with application to stability-encouraging identification  (2610.01295 - Rettberg et al., 1 Oct 2026) in Section 3.1, immediately before Theorem 3.5 (Theorem \ref{thm:OpInfNoConv})