On sharp frame diagonalization

It was recently shown that if M=Γ1⊕⋯⊕Γk∈Cn×n is a Jordan matrix with k nontrivial Jordan blocks Γi, then M can be frame diagonalized by embedding M into a diagonalizable matrix in C(n+ℓ)×(n+ℓ) with ℓ=k. This naturally motivates a pursuit of the best possible value of ℓ for which this is possible. Here, we use Lidskii’s Theorem on eigenvalue perturbations to construct diagonalizing frames for ℓ=k:max{gmM(λ)|λ∈σ(M)}. Moreover, we verify that k is sharp.