feat: port LinearAlgebra.Eigenspace.Minpoly (#4861)

Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>
Co-authored-by: Moritz Firsching <firsching@google.com>

Author: mo271

Mathlib.lean

@@ -1932,6 +1932,7 @@ import Mathlib.LinearAlgebra.DirectSum.TensorProduct
 import Mathlib.LinearAlgebra.Dual
 import Mathlib.LinearAlgebra.Eigenspace.Basic
 import Mathlib.LinearAlgebra.Eigenspace.IsAlgClosed
+import Mathlib.LinearAlgebra.Eigenspace.Minpoly
 import Mathlib.LinearAlgebra.FiniteDimensional
 import Mathlib.LinearAlgebra.Finrank
 import Mathlib.LinearAlgebra.Finsupp

Mathlib/Algebra/CharZero/Lemmas.lean

@@ -184,6 +184,15 @@ instance {R : Type _} [AddMonoidWithOne R] [CharZero R] :
 
 end WithTop
 
+namespace WithBot
+
+instance {R : Type _} [AddMonoidWithOne R] [CharZero R] :
+    CharZero (WithBot R) where
+  cast_injective m n h := by
+    rwa [← coe_nat, ← coe_nat n, coe_eq_coe, Nat.cast_inj] at h
+
+end WithBot
+
 section RingHom
 
 variable {R S : Type _} [NonAssocSemiring R] [NonAssocSemiring S]

Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean

@@ -0,0 +1,126 @@
+/-
+Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Alexander Bentkamp
+
+! This file was ported from Lean 3 source module linear_algebra.eigenspace.minpoly
+! leanprover-community/mathlib commit 6b0169218d01f2837d79ea2784882009a0da1aa1
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.LinearAlgebra.Eigenspace.Basic
+import Mathlib.FieldTheory.Minpoly.Field
+
+/-!
+# Eigenvalues are the roots of the minimal polynomial.
+
+## Tags
+
+eigenvalue, minimal polynomial
+-/
+
+
+universe u v w
+
+namespace Module
+
+namespace End
+
+open Polynomial FiniteDimensional
+
+open scoped Polynomial
+
+variable {K : Type v} {V : Type w} [Field K] [AddCommGroup V] [Module K V]
+
+theorem eigenspace_aeval_polynomial_degree_1 (f : End K V) (q : K[X]) (hq : degree q = 1) :
+    eigenspace f (-q.coeff 0 / q.leadingCoeff) = LinearMap.ker (aeval f q) :=
+  calc
+    eigenspace f (-q.coeff 0 / q.leadingCoeff)
+    _ = LinearMap.ker (q.leadingCoeff • f - algebraMap K (End K V) (-q.coeff 0)) := by
+          rw [eigenspace_div]
+          intro h
+          rw [leadingCoeff_eq_zero_iff_deg_eq_bot.1 h] at hq
+          cases hq
+    _ = LinearMap.ker (aeval f (C q.leadingCoeff * X + C (q.coeff 0))) := by
+          rw [C_mul', aeval_def]; simp [algebraMap, Algebra.toRingHom]
+    _ = LinearMap.ker (aeval f q) := by rwa [← eq_X_add_C_of_degree_eq_one]
+#align module.End.eigenspace_aeval_polynomial_degree_1 Module.End.eigenspace_aeval_polynomial_degree_1
+
+theorem ker_aeval_ring_hom'_unit_polynomial (f : End K V) (c : K[X]ˣ) :
+    LinearMap.ker (aeval f (c : K[X])) = ⊥ := by
+  rw [Polynomial.eq_C_of_degree_eq_zero (degree_coe_units c)]
+  simp only [aeval_def, eval₂_C]
+  apply ker_algebraMap_end
+  apply coeff_coe_units_zero_ne_zero c
+#align module.End.ker_aeval_ring_hom'_unit_polynomial Module.End.ker_aeval_ring_hom'_unit_polynomial
+
+theorem aeval_apply_of_hasEigenvector {f : End K V} {p : K[X]} {μ : K} {x : V}
+    (h : f.HasEigenvector μ x) : aeval f p x = p.eval μ • x := by
+  refine' p.induction_on _ _ _
+  · intro a; simp [Module.algebraMap_end_apply]
+  · intro p q hp hq; simp [hp, hq, add_smul]
+  · intro n a hna
+    rw [mul_comm, pow_succ, mul_assoc, AlgHom.map_mul, LinearMap.mul_apply, mul_comm, hna]
+    simp only [mem_eigenspace_iff.1 h.1, smul_smul, aeval_X, eval_mul, eval_C, eval_pow, eval_X,
+      LinearMap.map_smulₛₗ, RingHom.id_apply, mul_comm]
+#align module.End.aeval_apply_of_has_eigenvector Module.End.aeval_apply_of_hasEigenvector
+
+theorem isRoot_of_hasEigenvalue {f : End K V} {μ : K} (h : f.HasEigenvalue μ) :
+    (minpoly K f).IsRoot μ := by
+  rcases(Submodule.ne_bot_iff _).1 h with ⟨w, ⟨H, ne0⟩⟩
+  refine' Or.resolve_right (smul_eq_zero.1 _) ne0
+  simp [← aeval_apply_of_hasEigenvector ⟨H, ne0⟩, minpoly.aeval K f]
+#align module.End.is_root_of_has_eigenvalue Module.End.isRoot_of_hasEigenvalue
+
+variable [FiniteDimensional K V] (f : End K V)
+
+variable {f} {μ : K}
+
+theorem hasEigenvalue_of_isRoot (h : (minpoly K f).IsRoot μ) : f.HasEigenvalue μ := by
+  cases' dvd_iff_isRoot.2 h with p hp
+  rw [HasEigenvalue, eigenspace]
+  intro con
+  cases' (LinearMap.isUnit_iff_ker_eq_bot _).2 con with u hu
+  have p_ne_0 : p ≠ 0 := by
+    intro con
+    apply minpoly.ne_zero f.isIntegral
+    rw [hp, con, MulZeroClass.mul_zero]
+  have : (aeval f) p = 0 := by
+    have h_aeval := minpoly.aeval K f
+    revert h_aeval
+    simp [hp, ← hu]
+  have h_deg := minpoly.degree_le_of_ne_zero K f p_ne_0 this
+  rw [hp, degree_mul, degree_X_sub_C, Polynomial.degree_eq_natDegree p_ne_0] at h_deg
+  norm_cast at h_deg
+  linarith
+#align module.End.has_eigenvalue_of_is_root Module.End.hasEigenvalue_of_isRoot
+
+theorem hasEigenvalue_iff_isRoot : f.HasEigenvalue μ ↔ (minpoly K f).IsRoot μ :=
+  ⟨isRoot_of_hasEigenvalue, hasEigenvalue_of_isRoot⟩
+#align module.End.has_eigenvalue_iff_is_root Module.End.hasEigenvalue_iff_isRoot
+
+/-- An endomorphism of a finite-dimensional vector space has finitely many eigenvalues. -/
+noncomputable instance (f : End K V) : Fintype f.Eigenvalues :=
+  -- Porting note: added `show` to avoid unfolding `Set K` to `K → Prop`
+  show Fintype { μ : K | f.HasEigenvalue μ } from
+  Set.Finite.fintype
+    (by
+      have h : minpoly K f ≠ 0 := minpoly.ne_zero f.isIntegral
+      convert (minpoly K f).rootSet_finite K
+      ext μ
+      -- Porting note: was
+      -- have : μ ∈ {μ : K | f.eigenspace μ = ⊥ → False} ↔ ¬f.eigenspace μ = ⊥ := by tauto
+      -- convert rfl.mpr this
+      -- simp only [Polynomial.rootSet_def, Polynomial.mem_roots h, ← hasEigenvalue_iff_isRoot,
+      --   HasEigenvalue]
+      -- which didn't work, but worked with
+      -- simp only [Polynomial.rootSet_def, Polynomial.mem_roots h, ← hasEigenvalue_iff_isRoot,
+      --   HasEigenvalue, (Multiset.mem_toFinset), Algebra.id.map_eq_id, iff_self, Ne.def,
+      --   Polynomial.map_id, Finset.mem_coe]
+      -- but the code below is simpler.
+      rw [Set.mem_setOf_eq, hasEigenvalue_iff_isRoot, mem_rootSet_of_ne h, IsRoot,
+        coe_aeval_eq_eval])
+
+end End
+
+end Module