Eq 12 and Eq 18: Determinant and Trace are Product and Sum of Eigenvalues #10
Conversation
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Owner
|
Please merge master to resolve the conflicts |
MohanadAhmed
force-pushed
the
pr10_eigsLib
branch
from
4c6a767 to
a4f683c
Compare
eric-wieser
reviewed
Jun 8, 2023
src/matrix_cookbook/lib/eigs.lean
Outdated
Comment on lines
38
to
123
| lemma root_charpoly_of_has_eigenvalue (A: matrix n n R)(μ: R): | ||
| has_eigenvalue (matrix.to_lin' A) μ → A.charpoly.is_root μ:= | ||
| begin | ||
| intro heig, | ||
| have va := has_eigenvalue.exists_has_eigenvector heig, | ||
| have xa : (∃ (v : n → R) (H : v ≠ 0), (μ • 1 - A).mul_vec v = 0), | ||
| { cases va with v hv, | ||
| use v, | ||
| rw [has_eigenvector, mem_eigenspace_iff,to_lin'_apply, and_comm, eq_comm] at hv, | ||
| rwa [sub_mul_vec, smul_mul_vec_assoc, one_mul_vec, sub_eq_zero], }, | ||
| rw [matrix.charpoly, is_root, eval_det, mat_poly_equiv_charmatrix, | ||
| eval_sub, eval_C, eval_X, coe_scalar,← (matrix.exists_mul_vec_eq_zero_iff.1 xa)], | ||
| end | ||
|
|
||
| lemma has_eigenvalue_of_root_charpoly (A: matrix n n R)(μ: R): | ||
| A.charpoly.is_root μ → has_eigenvalue (matrix.to_lin' A) μ := | ||
| begin | ||
| rw [matrix.charpoly, is_root, eval_det, mat_poly_equiv_charmatrix,eval_sub], | ||
| rw [eval_C, eval_X, coe_scalar], | ||
| dsimp, | ||
|
|
||
| intro h, | ||
| have h := matrix.exists_mul_vec_eq_zero_iff.2 h, | ||
| rcases h with ⟨v, hv_nz, hv⟩, | ||
| rw [sub_mul_vec, smul_mul_vec_assoc, one_mul_vec, sub_eq_zero, eq_comm] at hv, | ||
| rw [has_eigenvalue, submodule.ne_bot_iff], | ||
| use v, | ||
| rw [mem_eigenspace_iff, to_lin'_apply], | ||
| split, assumption', | ||
| end | ||
|
|
||
| lemma root_charpoly_iff_has_eigenvalue (A: matrix n n R)(μ: R) : | ||
| A.charpoly.is_root μ ↔ has_eigenvalue (matrix.to_lin' A) μ := | ||
| begin | ||
| split, | ||
| apply has_eigenvalue_of_root_charpoly, | ||
| apply root_charpoly_of_has_eigenvalue, | ||
| end | ||
|
|
||
| lemma root_charpoly_iff_root_minpoly (A: matrix n n R)(μ: R) : | ||
| (minpoly R (matrix.to_lin' A)).is_root μ ↔ A.charpoly.is_root μ := | ||
| begin | ||
| split, | ||
| /- Bridge over has_eigenvalue to connect root_chapoly and root minpoly linear map-/ | ||
| intro h, | ||
| rw root_charpoly_iff_has_eigenvalue, | ||
| apply has_eigenvalue_iff_is_root.2 h, | ||
|
|
||
| rw root_charpoly_iff_has_eigenvalue, | ||
| intro h, | ||
| apply has_eigenvalue_iff_is_root.1 h, | ||
| end | ||
|
|
||
| lemma eig_if_eigenvalue (A: matrix n n R)(μ: R) : | ||
| μ ∈ eigs A → has_eigenvalue (matrix.to_lin' A) μ := | ||
| begin | ||
| rw eigs, | ||
| rw mem_roots', | ||
| intro h, | ||
| cases h with hne hcp, | ||
| exact has_eigenvalue_of_root_charpoly _ _ hcp, | ||
| end | ||
|
|
||
| lemma eigenvalue_if_eig [nonempty n](A: matrix n n R)(μ: R) : | ||
| has_eigenvalue (matrix.to_lin' A) μ → μ ∈ eigs A := | ||
| begin | ||
| rw eigs, rw mem_roots, | ||
| intro h, | ||
| exact root_charpoly_of_has_eigenvalue _ _ h, | ||
| have p_nz : matrix.charpoly A ≠ 0, | ||
| { by_contra, | ||
| replace h := congr_arg nat_degree h, | ||
| have p_deg := matrix.charpoly_nat_degree_eq_dim A, | ||
| have hn: fintype.card n ≠ 0, {exact fintype.card_ne_zero,}, | ||
| rw [nat_degree_zero, p_deg] at h, | ||
| exact hn h, }, | ||
| exact p_nz, | ||
| end | ||
|
|
||
| lemma mem_eigs_iff_eigenvalue [nonempty n] (A: matrix n n R)(μ: R): | ||
| μ ∈ eigs A ↔ has_eigenvalue (matrix.to_lin' A) μ := | ||
| begin | ||
| split, | ||
| apply eig_if_eigenvalue, | ||
| apply eigenvalue_if_eig, | ||
| end |
Owner
There was a problem hiding this comment.
You should PR these straight to mathlib! They're not actually used in the cookbook, right?
Contributor
Author
There was a problem hiding this comment.
The last two are related to the definition eig which is only here in the matrix cookbook. For the others I think you are probably right. Where do you think I should put them?
Owner
There was a problem hiding this comment.
I pushed a handful of golfs; even if this code goes straight to mathlib, we'll want them there. Hopefully seeing them is useful to you too!
Contributor
Author
There was a problem hiding this comment.
In which file in mathlib do you think they are suitable?
eric-wieser
added
the
lean3
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
2 participants
Add this suggestion to a batch that can be applied as a single commit.
This suggestion is invalid because no changes were made to the code.
Suggestions cannot be applied while the pull request is closed.
Suggestions cannot be applied while viewing a subset of changes.
Only one suggestion per line can be applied in a batch.
Add this suggestion to a batch that can be applied as a single commit.
Applying suggestions on deleted lines is not supported.
You must change the existing code in this line in order to create a valid suggestion.
Outdated suggestions cannot be applied.
This suggestion has been applied or marked resolved.
Suggestions cannot be applied from pending reviews.
Suggestions cannot be applied on multi-line comments.
Suggestions cannot be applied while the pull request is queued to merge.
Suggestion cannot be applied right now. Please check back later.
Determinant and Trace are Product and Sum of Eigenvalues
The eigenvalue as defined here is the same as the eigenvalue defined by
mathlibfor endomorphisms.root_charpoly_iff_has_eigenvalueand lemmas for both directionsmem_eigs_iff_eigenvalueand lemmas for both directions