Analogue of the Lyapunov spectral identity for genuinely coupled higher-rank updates

Determine whether genuinely coupled higher-rank data admit an analogue of the rank-one Lyapunov spectral identity under an appropriate matrix-valued Lyapunov-kernel or all-pass hypothesis.

Background

The paper proves a spectral identity for a rank-one state-matrix update of the form A_2=A+vw^\top, using a scalar Lyapunov-kernel factorization and product reversal. The authors explain that the unrestricted multicolumn extension is false, citing higher-rank counterexamples when v and w are replaced by matrix factors V and W with more than one column.

They note that simultaneous block reduction to independent rank-one channels preserves the identity blockwise. The unresolved case is therefore the genuinely coupled higher-rank setting, where the scalar kernel would need to be replaced or generalized by a matrix-valued Lyapunov-kernel or all-pass construction.