Expository Random-Matrix Framing Question

Definitions and mathematical statement

For terminology only, let $A\in K^{n\times n}$, where $K$ is a field, and let $L$ be an extension field of $K$. An eigenvalue of $A$ in $L$ is a scalar $\lambda\in L$ for which there exists $v\in L^n\setminus\{0\}$ satisfying \[ Av=\lambda v. \] Such a vector $v$ is an eigenvector, and the corresponding eigenspace is $\ker(A-\lambda I_n)$. These definitions supply vocabulary only; they do not specify the matrix ensemble intended by the source.

The retained source question is: What do the eigenvalues and eigenvectors of a typical large random matrix look like?

Unresolved formulation. The supplied statement-verification record identifies this sentence as a rhetorical introduction to a survey of established spectral and eigenvector universality results. It records the statement as unrecoverable: no unresolved conjecture, matrix ensemble, scale, or target theorem is specified, and no separate clean statement is supplied. In particular, the source does not determine the underlying field, the probability distributions of the matrices, structural or moment assumptions, the meaning of "typical," a limiting regime, spectral normalization, eigenvector normalization, observables, or a mode of convergence. Consequently, no precise quantified conjecture can be recovered from this record without additional choices. The registry's supplied open label does not by itself identify an unresolved mathematical proposition.

Short English statement

The record asks broadly about eigenvalues and eigenvectors of large random matrices. Its supplied verification identifies an expository framing question, with no specified ensemble or conjecture. A precise mathematical problem cannot be extracted without additional hypotheses and a target conclusion; the supplied open label requires this qualification.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30001802$, title Expository Random-Matrix Framing Question, source code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year 2011 and names no individual proposer.
• Supplied report citation: Stochastic Analysis (2011), Oberwolfach Reports, pp. 1573--1635, DOI 10.4171/owr/2011/29. https://doi.org/10.4171/owr/2011/29. This is also the supplied literature source labeled "original Oberwolfach report." No narrower passage location is supplied.
• Supplied literature source labeled "Erdos--Yau--Yin bulk universality": https://arxiv.org/abs/1001.3453.
• Supplied dated literature curation, checked 2026-08-21, labels the record open. Its assessment says that the broad question is not a single well-posed universality claim; it reports bulk and edge laws, rigidity, and eigenvector delocalization for broad Wigner classes, while stating that no general notion of a typical large matrix resolves the extracted question. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• Supplied statement-verification note: the question introduces a survey of established spectral and eigenvector universality results, rather than specifying an unresolved conjecture. The original record is retained for provenance, with statement status unrecoverable and no promoted clean statement.