A Test Matrix for an Inverse Eigenvalue Problem
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Bibliographic record
Abstract
We present a real symmetric tridiagonal matrix of order<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math>whose eigenvalues are<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msubsup><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math>which also satisfies the additional condition that its leading principle submatrix has a uniformly interlaced spectrum,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:msubsup><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>2</mml:mn><mml:mi>l</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow><mml:mrow><mml:mi>l</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math>. The matrix entries are explicit functions of the size<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math>, and so the matrix can be used as a test matrix for eigenproblems, both forward and inverse. An explicit solution of a spring-mass inverse problem incorporating the test matrix is provided.
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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 | 0.004 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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