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Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Characteristic polynomials and the Cayley-Hamilton theorem #

We define characteristic polynomials of matrices and prove the Cayley–Hamilton theorem over arbitrary commutative rings.

See the file Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean for corollaries of this theorem.

Main definitions #

Implementation details #

We follow a nice proof from http://drorbn.net/AcademicPensieve/2015-12/CayleyHamilton.pdf

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def Matrix.charmatrix {R : Type u_1} [CommRing R] {n : Type u_4} [DecidableEq n] [Fintype n] (M : Matrix n n R) :
Matrix n n (Polynomial R)

The "characteristic matrix" of M : Matrix n n R is the matrix of polynomials $t I - M$. The determinant of this matrix is the characteristic polynomial.

Equations
Instances For
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    theorem Matrix.charmatrix_apply {R : Type u_1} [CommRing R] {n : Type u_4} [DecidableEq n] [Fintype n] (M : Matrix n n R) (i : n) (j : n) :
    M.charmatrix i j = Matrix.diagonal (fun (x : n) => Polynomial.X) i j - Polynomial.C (M i j)
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    @[simp]
    theorem Matrix.charmatrix_apply_eq {R : Type u_1} [CommRing R] {n : Type u_4} [DecidableEq n] [Fintype n] (M : Matrix n n R) (i : n) :
    M.charmatrix i i = Polynomial.X - Polynomial.C (M i i)
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    @[simp]
    theorem Matrix.charmatrix_apply_ne {R : Type u_1} [CommRing R] {n : Type u_4} [DecidableEq n] [Fintype n] (M : Matrix n n R) (i : n) (j : n) (h : i ≠ j) :
    M.charmatrix i j = -Polynomial.C (M i j)
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    theorem Matrix.matPolyEquiv_charmatrix {R : Type u_1} [CommRing R] {n : Type u_4} [DecidableEq n] [Fintype n] (M : Matrix n n R) :
    matPolyEquiv M.charmatrix = Polynomial.X - Polynomial.C M
    source
    theorem Matrix.charmatrix_reindex {R : Type u_1} [CommRing R] {m : Type u_3} {n : Type u_4} [DecidableEq m] [DecidableEq n] [Fintype m] [Fintype n] (M : Matrix n n R) (e : n ≃ m) :
    ((Matrix.reindex e e) M).charmatrix = (Matrix.reindex e e) M.charmatrix
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    theorem Matrix.charmatrix_map {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {n : Type u_4} [DecidableEq n] [Fintype n] (M : Matrix n n R) (f : R →+* S) :
    (M.map ⇑f).charmatrix = M.charmatrix.map (Polynomial.map f)
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    theorem Matrix.charmatrix_fromBlocks {R : Type u_1} [CommRing R] {m : Type u_3} {n : Type u_4} [DecidableEq m] [DecidableEq n] [Fintype m] [Fintype n] (M₁₁ : Matrix m m R) (M₁₂ : Matrix m n R) (M₂₁ : Matrix n m R) (M₂₂ : Matrix n n R) :
    (M₁₁.fromBlocks M₁₂ M₂₁ M₂₂).charmatrix = M₁₁.charmatrix.fromBlocks (-M₁₂.map ⇑Polynomial.C) (-M₂₁.map ⇑Polynomial.C) M₂₂.charmatrix
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    def Matrix.charpoly {R : Type u_1} [CommRing R] {n : Type u_4} [DecidableEq n] [Fintype n] (M : Matrix n n R) :
    Polynomial R

    The characteristic polynomial of a matrix M is given by $\det (t I - M)$.

    Equations
    • M.charpoly = M.charmatrix.det
    Instances For
      source
      theorem Matrix.charpoly_reindex {R : Type u_1} [CommRing R] {m : Type u_3} {n : Type u_4} [DecidableEq m] [DecidableEq n] [Fintype m] [Fintype n] (e : n ≃ m) (M : Matrix n n R) :
      ((Matrix.reindex e e) M).charpoly = M.charpoly
      source
      theorem Matrix.charpoly_map {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {n : Type u_4} [DecidableEq n] [Fintype n] (M : Matrix n n R) (f : R →+* S) :
      (M.map ⇑f).charpoly = Polynomial.map f M.charpoly
      source
      @[simp]
      theorem Matrix.charpoly_fromBlocks_zero₁₂ {R : Type u_1} [CommRing R] {m : Type u_3} {n : Type u_4} [DecidableEq m] [DecidableEq n] [Fintype m] [Fintype n] (M₁₁ : Matrix m m R) (M₂₁ : Matrix n m R) (M₂₂ : Matrix n n R) :
      (M₁₁.fromBlocks 0 M₂₁ M₂₂).charpoly = M₁₁.charpoly * M₂₂.charpoly
      source
      @[simp]
      theorem Matrix.charpoly_fromBlocks_zero₂₁ {R : Type u_1} [CommRing R] {m : Type u_3} {n : Type u_4} [DecidableEq m] [DecidableEq n] [Fintype m] [Fintype n] (M₁₁ : Matrix m m R) (M₁₂ : Matrix m n R) (M₂₂ : Matrix n n R) :
      (M₁₁.fromBlocks M₁₂ 0 M₂₂).charpoly = M₁₁.charpoly * M₂₂.charpoly
      source
      theorem Matrix.aeval_self_charpoly {R : Type u_1} [CommRing R] {n : Type u_4} [DecidableEq n] [Fintype n] (M : Matrix n n R) :
      (Polynomial.aeval M) M.charpoly = 0

      The Cayley-Hamilton Theorem, that the characteristic polynomial of a matrix, applied to the matrix itself, is zero.

      This holds over any commutative ring.

      See LinearMap.aeval_self_charpoly for the equivalent statement about endomorphisms.