feat(LinearAlgebra/Matrix/Charpoly): general coefficient formula as sum of principal minors

[slavanaprienko]

This PR adds a general formula for the coefficients of det(1 + X • M) as sums of principal minors, generalizing the existing coeff_det_one_add_X_smul_one (k=1, trace) and det_eq_sign_charpoly_coeff (k=n, determinant). It also derives the corresponding formula for the coefficients of the characteristic polynomial via charpolyRev.

New results:

• det_piecewise_one_eq_submatrix_det: relates piecewise-with-identity to principal submatrix determinants
• coeff_det_one_add_X_smul_eq_sum_minors: the k-th coefficient of det(1 + X • M) equals the sum of k×k principal minors
• charpoly_coeff_eq_sum_minors: coefficients of the characteristic polynomial as signed sums of principal minors

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[github-actions]

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[github-actions]

PR summary f211ee9026

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff

+ charpoly_coeff_eq_sum_minors
+ coeff_det_one_add_X_smul_eq_sum_minors
+ det_piecewise_one_eq_submatrix_det

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[slavanaprienko]

Updated with all the review feedback. Removed the inline comments, switched to let +generalize, used M.row in piecewise, dot notation for powersetCard, and nontriviality instead of [Nontrivial R]. The missing AlternatingMap API and Matrix.map_neg/map_sub I'll address in follow-up PRs if it's okay.

[eric-wieser]

The missing AlternatingMap API and Matrix.map_neg/map_sub I'll address in follow-up PRs if it's okay.

Could you do these in a pre-work PR instead? That can hopefully be merged quickly (feel free to ping me), then you can merge master here.

[slavanaprienko]

The missing AlternatingMap API and Matrix.map_neg/map_sub I'll address in follow-up PRs if it's okay.

Could you do these in a pre-work PR instead? That can hopefully be merged quickly (feel free to ping me), then you can merge master here.

Here's the pre-work PR:
#37719

Adds AlternatingMap.map_add_univ, AlternatingMap.map_smul_univ, and Matrix.map_neg. Let me know if anything needs adjusting.

[slavanaprienko]

The missing AlternatingMap API and Matrix.map_neg/map_sub I'll address in follow-up PRs if it's okay.

Could you do these in a pre-work PR instead? That can hopefully be merged quickly (feel free to ping me), then you can merge master here.

Done. #37719 is merged. Now we use AlternatingMap.map_add_univ, map_smul_univ, and Matrix.map_neg directly.