Some comments on several matrix inequalities with applications to canonical correlations: Historical background and recent developments
Bibliographic record
Abstract
We review several matrix inequalities and give some statistical applications, with special emphasis on canonical correlations; many historical and biographical remarks are also included as well as over 100 references. Ourpaper builds upon the recent survey by Alpargu and Styan (2000) and concentrates on recent developments. We present a new Generalized Matrix Frucht-Kantorovich inequality and show that it is essentially equivalent to the Generalized Matrix Wielandt inequality given by Lu (1999), extending recent results by Wang and Ip (1999). We discuss an interesting special case involving block rank additivity of a partitioned matrix and offer several characterizations. We also consider the Krasnosel'skˇiˇi-Kreˇin inequality and the Shisha-Mond inequality and matrix extensions due to Khatri and Rao (1981, 1982) and Rao (1985). Some related inequalities involving determinants and traces are also presented. We establish some new inequalities and give a proof for an upper bound for the product of canonical correlations stated by Khatri (1982) and Khatri and Rao (1982). In addition, we present a new proof of the Bloomfield-Watson-Knott inequality; the Bloomfield-Watson-Knott, Khatri-Rao and Rao inequalities are identified as essential for exciting new results on majorization of eigenvalues due to Ando (2000, 2001) and Li and Mathias (1999).
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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 | 0.003 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.005 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.007 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.014 | 0.005 |
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".