Combinatorial considerations for the number of distinct eigenvalues of a matrix
Bibliographic record
Abstract
We address the inverse eigenvalue problem of determining the potential number of distinct eigenvalues of a real matrix based on the zero-nonzero structure of the matrix. In particular, a nonzero pattern $\mathcal{A}$ is a matrix with entries in $\{*,0\}$. The allow sequence of distinct eigenvalues for an $n\times n$ pattern $\mathcal{A}$ is a binary vector of length $n$ with the $k$th entry equal to $1$ if and only if there exists a real matrix with pattern $\mathcal{A}$ having exactly $k$ distinct eigenvalues. We develop digraph techniques for identifying properties of the allow sequence and give some general results for cycle patterns. We obtain a classification for all the star patterns according to their allow sequence. We also determine the allow sequence for each $n\times n$ irreducible pattern with $n\leq 4$.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".