A note on companion pencils
Bibliographic record
Abstract
Various generalizations of companion matrices to companion pencils are presented. Companion matrices link to monic polynomials, whereas companion pencils do not require monicity of the corresponding polynomial. In the classical companion pencil case ( A , B ) (A,B) only the coefficient of the highest degree appears in B B ’s lower right corner. We will show, however, that all coefficients of the polynomial can be distributed over both A A and B B creating additional flexibility. Companion matrices admit a so-called Fiedler factorization into essentially 2 × 2 2\times 2 matrices. These Fiedler factors can be reordered without affecting the eigenvalues (which equal the polynomial roots) of the assembled matrix. We will propose a generalization of this factorization and the reshuffling property for companion pencils. Special examples of the factorizations and extensions to matrix polynomials and product eigenvalue problems are included.
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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 | 0.004 | 0.022 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.005 | 0.012 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.048 | 0.023 |
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".