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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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A comma upper B right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(A,B)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> only the coefficient of the highest degree appears in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ’s lower right corner. We will show, however, that all coefficients of the polynomial can be distributed over both <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> creating additional flexibility. Companion matrices admit a so-called Fiedler factorization into essentially <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 times 2"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo> × </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2\times 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.004 |
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 it