Spectral properties of deterministic matrices multiplied by rotationally invariant random non-Hermitian ensembles

What the math gives to ML

The paper provides asymptotic spectral control for products of a deterministic matrix with a rotationally invariant non-Hermitian random matrix, a setting relevant to recurrent transition maps, state-space models, and deep linear Jacobians. Its transferable asset is a relation between the complex eigenvalue support of a product and the S-transforms of the squared-singular-value distributions of its factors. A practical use is to initialize or rescale recurrent operators so that their predicted eigenvalue cloud lies in a target stability annulus, directly targeting long-horizon dynamics rather than only singular-value normalization. The result is most credible as an initialization and diagnostic proxy when widths are large and one factor is approximately rotationally invariant.