[Merged by Bors] - feat(linear_algebra/cayley_hamilton): the Cayley-Hamilton theorem #3276
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I'm looking forward to someone constructing eigenvectors now! :-) |
Co-authored-by: Bryan Gin-ge Chen <bryangingechen@gmail.com>
…mathlib into polynomial_algebra
…unity/mathlib into cayley_hamilton_squashed
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Co-authored-by: Johan Commelin <johan@commelin.net>
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#3333) As discussed on [zulip](https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/import.20refactor.20and.20library_search), #3235 had the unfortunate effect of making `library_search` and `#where` and various other things unavailable in many places in mathlib. This PR makes an effort to import `tactic.basic` as early as possible, while otherwise reducing unnecessary imports. 1. import `tactic.basic` "as early as possible" (i.e. in any file that `tactic.basic` doesn't depend on, and which imports any tactic strictly between `tactic.core` and `tactic.basic`, just `import tactic.basic` itself 2. add `tactic.finish`, `tactic.tauto` and `tactic.norm_cast` to tactic.basic (doesn't requires adding any dependencies) 3. delete various unnecessary imports Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
Other changes: * rename `injective.prod` to `injective.prod_map`; * add `surjective.prod_map`; * redefine `sigma.map` without pattern matching; * rename `sigma_map_injective` to `injective.sigma_map`; * add `surjective.sigma_map`; * add `injective.sum_map` and `surjective.sum_map`; * rename `embedding.prod_congr` to `embedding.prod_map`; * rename `embedding.sum_congr` to `embedding.sum_map`; * delete `embedding.sigma_congr_right`, add more general `embedding.sigma_map`.
Remove the unnecessary assumption that `A` is commutative in `matrix n n A ≃ₐ[R] (A ⊗[R] matrix n n R)`. (This didn't cause a problem for Cayley-Hamilton, but @alexjbest and Bassem Safieldeen [realised it's not necessary](https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Tensor.20product.20of.20matrices).) Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
A refactor of `algebra/homology` so homology and cohomology are handled uniformly, and factor out a file `image_to_kernel_map.lean` which gains some extra lemmas (which will be useful for talking about exact sequences). Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Johan Commelin <johan@commelin.net>
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) The Cayley-Hamilton theorem, following the proof at http://drorbn.net/AcademicPensieve/2015-12/CayleyHamilton.pdf. Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Yury G. Kudryashov <urkud@urkud.name> Co-authored-by: Johan Commelin <johan@commelin.net>
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| -- is sent to zero, because the evaluation function puts the polynomial variable | ||
| -- to the right of any coefficients, so everything telescopes. | ||
| apply_fun (λ p, p.eval M) at h, | ||
| rw eval_mul_X_sub_C at h, |
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Is there any reason not to write rwa [eval_mul_X_sub_C, mat_poly_equiv_smul_one] at h, instead of those three lines? Is that because of the narrative in the comments?
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I think that this proof was indeed un-golfed for readability reasons.
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I wonder whether we should care about the fact that mathcomp people always point out how their proof is only three lines long.
Maybe we can add a comment below our proof saying:
theorem char_poly_map_eval_self (M : matrix n n R) :
((char_poly M).map (algebra_map R (matrix n n R))).eval M = 0 :=
begin
have h : (mat_poly_equiv ((char_poly M) • 1)).eval M = (mat_poly_equiv (adjugate (char_matrix M) * (char_matrix M))).eval M := congr_arg _ (congr_arg _ (adjugate_mul _).symm),
simpa only [mat_poly_equiv.map_mul, mat_poly_equiv_char_matrix, eval_mul_X_sub_C, mat_poly_equiv_smul_one] using h,
end
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Sounds good to me.
How about by simpa ... using ...?
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Indeed, on Scott's screen you could use
by simpa only [mat_poly_equiv.map_mul, mat_poly_equiv_char_matrix, eval_mul_X_sub_C, mat_poly_equiv_smul_one] using (congr_arg _ (congr_arg _ (adjugate_mul _).symm) : (mat_poly_equiv ((char_poly M) • 1)).eval M = (mat_poly_equiv (adjugate (char_matrix M) * (char_matrix M))).eval M)
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…anprover-community#3276) The Cayley-Hamilton theorem, following the proof at http://drorbn.net/AcademicPensieve/2015-12/CayleyHamilton.pdf. Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Yury G. Kudryashov <urkud@urkud.name> Co-authored-by: Johan Commelin <johan@commelin.net>
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The Cayley-Hamilton theorem, following the proof at http://drorbn.net/AcademicPensieve/2015-12/CayleyHamilton.pdf.
This PR is currently stacked on top of #3275, but the only new file here islinear_algebra/cayley_hamilton.leanif you want to see what it looks like.Now blocked by #3352, arising from review suggestions.