Linear Systems and Eigenvalue Problems — Problem 25

Definitions and mathematical statement

Let $n,k\in{\mathbb{N}}$ satisfy $1<k<n$, and let $A\in{\mathbb{C}}^{n\times n}$ be non-Hermitian, meaning $A\ne A^*$, where $^*$ denotes conjugate transpose. For $Q\in{\mathbb{C}}^{n\times k}$ with $Q^*Q=I_k$, define the compression $B_Q=Q^*AQ$. Its eigenvalues $\theta_1(Q),\ldots,\theta_k(Q)$, counted with algebraic multiplicity, are the Ritz values associated with $Q$. Define the numerical range by \[ W(A)=\{v^*Av:v\in{\mathbb{C}}^n,\ \|v\|_2=1\}. \] The supplied source recalls that $W(A)$ is closed and convex, contains the eigenvalues of $A$, and contains every Ritz value; it may also contain points far from the eigenvalues of $A$. To express the distribution of the Ritz values, one may use the empirical measure \[ \mu_{A,Q}=\frac1k\sum_{j=1}^k\delta_{\theta_j(Q)}, \] where $\delta_z$ is the unit point mass at $z$.

Source question. What deterministic or probabilistic information can be obtained about the distribution of these Ritz values across $W(A)$ when $1<k<n$? How does the question change when the columns of $Q$ form an orthonormal basis of a Krylov subspace, or an orthonormal basis of a random subspace?

For clarity, a standard single-vector Krylov subspace is \[ \mathcal K_k(A,b)=\operatorname{span}\{b,Ab,\ldots,A^{k-1}b\},\qquad b\in{\mathbb{C}}^n\setminus\{0\}. \] Under this interpretation, an orthonormal basis with exactly $k$ columns requires $\dim\mathcal K_k(A,b)=k$. The source does not specify the starting vector, whether this particular Krylov construction is intended, or how breakdown is to be handled. It also does not specify a probability law for a random subspace, any randomness in the Krylov construction, a particular distributional statistic or bound, or an asymptotic regime. Thus the supplied formulation is an open research question rather than a fully specified conjecture; its probabilistic variants require additional choices.

Short English statement

For a non-Hermitian matrix, describe how the eigenvalues of its compressions to subspaces of dimension $k$, with $1<k<n$, are distributed inside its numerical range. Investigate arbitrary subspaces, Krylov subspaces, and random subspaces. The source leaves the desired distributional conclusions, Krylov construction, and probability laws unspecified.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $800516$, source problem code catalogue entry, title "Linear Systems and Eigenvalue Problems --- Problem 25." The registry attributes the problem to Amsel, Baumann et al., with proposed year $2026$, and labels it open. This status is supplied metadata, not an independently verified mathematical or literature assessment.
• Amsel, Baumann et al., Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop (2026), Problem 25. Supplied source URL: https://arxiv.org/abs/2602.05394. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent source access.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.