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[Merged by Bors] - fix(analysis/inner_product_space/pi_L2): add missing type cast functions #18574
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fdc7ef5
fix(analysis/inner_product_space/pi_L2): add missing type cast functions
eric-wieser b317bca
better notation
eric-wieser becd619
Merge branch 'master' into eric-wieser/inner-pi_Lp-fixes
eric-wieser c44e248
add to_euclidean_lin
eric-wieser fbd0ec2
fix proof
eric-wieser da71675
lintfix
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Original file line number | Diff line number | Diff line change |
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@@ -25,7 +25,7 @@ variables {α β : Type*} {m n : Type*} {A : matrix n n α} | |
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open_locale matrix | ||
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local notation `⟪`x`, `y`⟫` := @inner α (pi_Lp 2 (λ (_ : n), α)) _ x y | ||
local notation `⟪`x`, `y`⟫` := @inner α _ _ x y | ||
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section non_unital_semiring | ||
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@@ -200,14 +200,11 @@ funext h.coe_re_apply_self | |
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/-- A matrix is hermitian iff the corresponding linear map is self adjoint. -/ | ||
lemma is_hermitian_iff_is_symmetric [fintype n] [decidable_eq n] {A : matrix n n α} : | ||
is_hermitian A ↔ linear_map.is_symmetric | ||
((pi_Lp.linear_equiv 2 α (λ _ : n, α)).symm.conj A.to_lin' : module.End α (pi_Lp 2 _)) := | ||
is_hermitian A ↔ A.to_euclidean_lin.is_symmetric := | ||
begin | ||
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There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. This spelling (suggested in #14231 (comment)) was already type-correct, but has been folded into |
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rw [linear_map.is_symmetric, (pi_Lp.equiv 2 (λ _ : n, α)).symm.surjective.forall₂], | ||
simp only [linear_equiv.conj_apply, linear_map.comp_apply, linear_equiv.coe_coe, | ||
pi_Lp.linear_equiv_apply, pi_Lp.linear_equiv_symm_apply, linear_equiv.symm_symm], | ||
simp_rw [euclidean_space.inner_eq_star_dot_product, equiv.apply_symm_apply, to_lin'_apply, | ||
star_mul_vec, dot_product_mul_vec], | ||
simp only [to_euclidean_lin_pi_Lp_equiv_symm, euclidean_space.inner_pi_Lp_equiv_symm, | ||
to_lin'_apply, star_mul_vec, dot_product_mul_vec], | ||
split, | ||
{ rintro (h : Aᴴ = A) x y, | ||
rw h }, | ||
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The statement of this lemma was the subject of some discussion in #11551 (comment), and in this Zulip thread.
Note the new version is now consistent with the statement of
matrix.is_hermitian_iff_is_symmetric
below.In the year since it was added, there are still no consumers of this lemma.