[Merged by Bors] - refactor(linear_algebra/eigenspace): refactor exists_eigenvalue #7345
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Vierkantor
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May 2, 2021
| @@ -56,13 +56,20 @@ def eigenspace (f : End R M) (ΞΌ : R) : submodule R M := | |||
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| /-- A nonzero element of an eigenspace is an eigenvector. (Def 5.7 of [axler2015]) -/ | |||
| def has_eigenvector (f : End R M) (ΞΌ : R) (x : M) : Prop := | |||
| x β 0 β§ x β eigenspace f ΞΌ | |||
| x β eigenspace f ΞΌ β§ x β 0 | |||
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I'm curious about the decision to switch this around. (Not that I have a reason to prefer it the old way.)
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It ... appears to be solely so that I can write
exact has_eigenvalue_of_has_eigenvector (exists_mem_ne_zero_of_ne_bot nu).some_spec
below, without having to flip over what I pull out of exists_mem_ne_zero_of_ne_bot.
One the one hand: why bother making this small change that affects several files? On the other hand: the changes required are all neutral, so why not?
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Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>
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We replace the proof of `exists_eigenvalue` with the more general fact:
```
/--
Every element `f` in a nontrivial finite-dimensional algebra `A`
over an algebraically closed field `K`
has non-empty spectrum:
that is, there is some `c : K` so `f - c β’ 1` is not invertible.
-/
lemma exists_spectrum_of_is_alg_closed_of_finite_dimensional (π : Type*) [field π] [is_alg_closed π]
{A : Type*} [nontrivial A] [ring A] [algebra π A] [I : finite_dimensional π A] (f : A) :
β c : π, Β¬ is_unit (f - algebra_map π A c) := ...
```
We can then use this fact to prove Schur's lemma.
Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
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We replace the proof of
exists_eigenvaluewith the more general fact:We can then use this fact to prove Schur's lemma.