Characteristic polynomials #
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We give methods for computing coefficients of the characteristic polynomial.
Main definitions #
matrix.charpoly_degree_eq_dimproves that the degree of the characteristic polynomial over a nonzero ring is the dimension of the matrixmatrix.det_eq_sign_charpoly_coeffproves that the determinant is the constant term of the characteristic polynomial, up to sign.matrix.trace_eq_neg_charpoly_coeffproves that the trace is the negative of the (d-1)th coefficient of the characteristic polynomial, where d is the dimension of the matrix. For a nonzero ring, this is the second-highest coefficient.
theorem
charmatrix_apply_nat_degree
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
{M : matrix n n R}
[nontrivial R]
(i j : n) :
(charmatrix M i j).nat_degree = ite (i = j) 1 0
theorem
charmatrix_apply_nat_degree_le
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
{M : matrix n n R}
(i j : n) :
(charmatrix M i j).nat_degree ≤ ite (i = j) 1 0
theorem
matrix.charpoly_sub_diagonal_degree_lt
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
(M : matrix n n R) :
(M.charpoly - finset.univ.prod (λ (i : n), polynomial.X - ⇑polynomial.C (M i i))).degree < ↑(fintype.card n - 1)
theorem
matrix.charpoly_coeff_eq_prod_coeff_of_le
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
(M : matrix n n R)
{k : ℕ}
(h : fintype.card n - 1 ≤ k) :
M.charpoly.coeff k = (finset.univ.prod (λ (i : n), polynomial.X - ⇑polynomial.C (M i i))).coeff k
theorem
matrix.det_of_card_zero
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
(h : fintype.card n = 0)
(M : matrix n n R) :
theorem
matrix.charpoly_degree_eq_dim
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
[nontrivial R]
(M : matrix n n R) :
M.charpoly.degree = ↑(fintype.card n)
theorem
matrix.charpoly_nat_degree_eq_dim
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
[nontrivial R]
(M : matrix n n R) :
theorem
matrix.charpoly_monic
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
(M : matrix n n R) :
theorem
matrix.mat_poly_equiv_eval
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
(M : matrix n n (polynomial R))
(r : R)
(i j : n) :
polynomial.eval (⇑(matrix.scalar n) r) (⇑mat_poly_equiv M) i j = polynomial.eval r (M i j)
theorem
matrix.eval_det
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
(M : matrix n n (polynomial R))
(r : R) :
polynomial.eval r M.det = (polynomial.eval (⇑(matrix.scalar n) r) (⇑mat_poly_equiv M)).det
theorem
matrix.det_eq_sign_charpoly_coeff
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
(M : matrix n n R) :
theorem
mat_poly_equiv_eq_X_pow_sub_C
{n : Type v}
[decidable_eq n]
[fintype n]
{K : Type u_1}
(k : ℕ)
[field K]
(M : matrix n n K) :
⇑mat_poly_equiv (⇑(↑(polynomial.expand K k).map_matrix) (charmatrix (M ^ k))) = polynomial.X ^ k - ⇑polynomial.C (M ^ k)
theorem
matrix.aeval_eq_aeval_mod_charpoly
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
(M : matrix n n R)
(p : polynomial R) :
⇑(polynomial.aeval M) p = ⇑(polynomial.aeval M) (p %ₘ M.charpoly)
Any matrix polynomial p is equivalent under evaluation to p %ₘ M.charpoly; that is, p
is equivalent to a polynomial with degree less than the dimension of the matrix.
theorem
matrix.pow_eq_aeval_mod_charpoly
{R : Type u}
[comm_ring R]
{n : Type v}
[decidable_eq n]
[fintype n]
(M : matrix n n R)
(k : ℕ) :
M ^ k = ⇑(polynomial.aeval M) (polynomial.X ^ k %ₘ M.charpoly)
Any matrix power can be computed as the sum of matrix powers less than fintype.card n.
TODO: add the statement for negative powers phrased with zpow.