/- Copyright (c) 2021 Anne Baanen. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Anne Baanen -/ import data.matrix.basic import linear_algebra.matrix.determinant import linear_algebra.matrix.adjugate /-! # Matrices associated with non-degenerate bilinear forms ## Main definitions * `matrix.nondegenerate A`: the proposition that when interpreted as a bilinear form, the matrix `A` is nondegenerate. -/ namespace matrix variables {m R A : Type*} [fintype m] [comm_ring R] /-- A matrix `M` is nondegenerate if for all `v ≠ 0`, there is a `w ≠ 0` with `w ⬝ M ⬝ v ≠ 0`. -/ def nondegenerate (M : matrix m m R) := ∀ v, (∀ w, matrix.dot_product v (mul_vec M w) = 0) → v = 0 /-- If `M` is nondegenerate and `w ⬝ M ⬝ v = 0` for all `w`, then `v = 0`. -/ lemma nondegenerate.eq_zero_of_ortho {M : matrix m m R} (hM : nondegenerate M) {v : m → R} (hv : ∀ w, matrix.dot_product v (mul_vec M w) = 0) : v = 0 := hM v hv /-- If `M` is nondegenerate and `v ≠ 0`, then there is some `w` such that `w ⬝ M ⬝ v ≠ 0`. -/ lemma nondegenerate.exists_not_ortho_of_ne_zero {M : matrix m m R} (hM : nondegenerate M) {v : m → R} (hv : v ≠ 0) : ∃ w, matrix.dot_product v (mul_vec M w) ≠ 0 := not_forall.mp (mt hM.eq_zero_of_ortho hv) variables [comm_ring A] [is_domain A] /-- If `M` has a nonzero determinant, then `M` as a bilinear form on `n → A` is nondegenerate. See also `bilin_form.nondegenerate_of_det_ne_zero'` and `bilin_form.nondegenerate_of_det_ne_zero`. -/ theorem nondegenerate_of_det_ne_zero [decidable_eq m] {M : matrix m m A} (hM : M.det ≠ 0) : nondegenerate M := begin intros v hv, ext i, specialize hv (M.cramer (pi.single i 1)), refine (mul_eq_zero.mp _).resolve_right hM, convert hv, simp only [mul_vec_cramer M (pi.single i 1), dot_product, pi.smul_apply, smul_eq_mul], rw [finset.sum_eq_single i, pi.single_eq_same, mul_one], { intros j _ hj, simp [hj] }, { intros, have := finset.mem_univ i, contradiction } end theorem eq_zero_of_vec_mul_eq_zero [decidable_eq m] {M : matrix m m A} (hM : M.det ≠ 0) {v : m → A} (hv : M.vec_mul v = 0) : v = 0 := (nondegenerate_of_det_ne_zero hM).eq_zero_of_ortho (λ w, by rw [dot_product_mul_vec, hv, zero_dot_product]) theorem eq_zero_of_mul_vec_eq_zero [decidable_eq m] {M : matrix m m A} (hM : M.det ≠ 0) {v : m → A} (hv : M.mul_vec v = 0) : v = 0 := eq_zero_of_vec_mul_eq_zero (by rwa det_transpose) ((vec_mul_transpose M v).trans hv) end matrix