Conditioning and Pseudospectral Size of Ritz Matrices from Krylov Spaces

Ascertain whether, for a Krylov subspace basis Q generated from a random starting vector b and an arbitrary matrix A ∈ C^{n×n}, the eigenvector condition number κ_V(Q^*AQ) is polynomially bounded in n with high probability; alternatively, determine whether the expected area of the ε-pseudospectrum of Q^*AQ is bounded by poly(n)^β for β close to 2.

Background

Accurate Ritz value approximations rely not only on eigenvalue inclusion but also on the conditioning and pseudospectral size of the projected matrix Q*AQ.

Pathological starting vectors can yield highly nonnormal or even defective projected matrices (e.g., Jordan blocks), but such cases may be rare under random starts; formalizing this intuition remains open.

References

Let $Q$ be an orthonormal basis for a Krylov space for arbitrary $A\in{n\times n}$ with random starting vector $b$. Is $\kappa_V(Q*AQ)$ bounded (by a polynomial in $n$) with high probability? Or is $[\area\Lambda_(Q*AQ)]$ (the expected value of the area in $$ of the $\varepsilon$-pseudospectrum) bounded by $poly(n)\beta$ for $\beta$ close to 2? (This problem is open even when $A$ is a circulant shift matrix.)

— Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop  (2602.05394 - Amsel et al., 5 Feb 2026) in Subsection "Ritz values and approximate invariant subspaces" (Section 3)