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.

def Matrix.SeparatingRight {m : Type u_1} {n : Type u_2} {R : Type u_3} [CommRing R] [Finite m] [Finite n] (M : Matrix m n R) : Prop

A matrix M is right-separating if for all w ≠ 0, there is a v with v * M * w ≠ 0.

def Matrix.SeparatingLeft {m : Type u_1} {n : Type u_2} {R : Type u_3} [CommRing R] [Finite m] [Finite n] (M : Matrix m n R) : Prop

A matrix M is right-separating if for all v ≠ 0, there is a w with v * M * w ≠ 0.

structure Matrix.Nondegenerate {m : Type u_1} {n : Type u_2} {R : Type u_3} [CommRing R] [Finite m] [Finite n] (M : Matrix m n R) : Prop

A matrix M is nondegenerate if it is both left-separating and right-separating.

• separatingLeft : M.SeparatingLeft
• separatingRight : M.SeparatingRight

theorem Matrix.separatingRight_def {m : Type u_1} {n : Type u_2} {R : Type u_3} [CommRing R] [Fintype m] [Fintype n] {M : Matrix m n R} : M.SeparatingRight ↔ ∀ (w : n → R), (∀ (v : m → R), v ⬝ᵥ M.mulVec w = 0) → w = 0

theorem Matrix.separatingLeft_def {m : Type u_1} {n : Type u_2} {R : Type u_3} [CommRing R] [Fintype m] [Fintype n] {M : Matrix m n R} : M.SeparatingLeft ↔ ∀ (v : m → R), (∀ (w : n → R), v ⬝ᵥ M.mulVec w = 0) → v = 0

theorem Matrix.nondegenerate_def {m : Type u_1} {n : Type u_2} {R : Type u_3} [CommRing R] [Fintype m] [Fintype n] {M : Matrix m n R} : M.Nondegenerate ↔ (∀ (v : m → R), (∀ (w : n → R), v ⬝ᵥ M.mulVec w = 0) → v = 0) ∧ ∀ (w : n → R), (∀ (v : m → R), v ⬝ᵥ M.mulVec w = 0) → w = 0

theorem Matrix.Nondegenerate.eq_zero_of_ortho {m : Type u_1} {n : Type u_2} {R : Type u_3} [CommRing R] [Fintype m] [Fintype n] {M : Matrix m n R} (hM : M.Nondegenerate) {v : m → R} (hv : ∀ (w : n → R), v ⬝ᵥ M.mulVec w = 0) : v = 0

If M is nondegenerate and w * M * v = 0 for all w, then v = 0.

theorem Matrix.Nondegenerate.exists_not_ortho_of_ne_zero {m : Type u_1} {n : Type u_2} {R : Type u_3} [CommRing R] [Fintype m] [Fintype n] {M : Matrix m n R} (hM : M.Nondegenerate) {v : m → R} (hv : v ≠ 0) : ∃ (w : n → R), v ⬝ᵥ M.mulVec w ≠ 0

If M is nondegenerate and v ≠ 0, then there is some w such that w * M * v ≠ 0.

theorem Matrix.Nondegenerate.eq_zero_of_ortho' {m : Type u_1} {n : Type u_2} {R : Type u_3} [CommRing R] [Fintype m] [Fintype n] {M : Matrix m n R} (hM : M.Nondegenerate) {w : n → R} (hw : ∀ (v : m → R), v ⬝ᵥ M.mulVec w = 0) : w = 0

If M is nondegenerate and w * M * v = 0 for all v, then w = 0.

theorem Matrix.Nondegenerate.exists_not_ortho_of_ne_zero' {m : Type u_1} {n : Type u_2} {R : Type u_3} [CommRing R] [Fintype m] [Fintype n] {M : Matrix m n R} (hM : M.Nondegenerate) {w : n → R} (hw : w ≠ 0) : ∃ (v : m → R), v ⬝ᵥ M.mulVec w ≠ 0

If M is nondegenerate and w ≠ 0, then there is some v such that v * M * w ≠ 0.

theorem Matrix.nondegenerate_of_det_ne_zero {m : Type u_1} {A : Type u_4} [Fintype m] [CommRing A] [IsDomain A] [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0) : M.Nondegenerate

If M is square and has nonzero determinant, then M as a bilinear form on n → A is nondegenerate. The "iff" implication, nondegenerate_iff_det_ne_zero, is proved in a later file.

See also BilinForm.nondegenerateOfDetNeZero' and BilinForm.nondegenerateOfDetNeZero.

theorem Matrix.eq_zero_of_vecMul_eq_zero {m : Type u_1} {A : Type u_4} [Fintype m] [CommRing A] [IsDomain A] [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0) {v : m → A} (hv : vecMul v M = 0) : v = 0

theorem Matrix.eq_zero_of_mulVec_eq_zero {m : Type u_1} {A : Type u_4} [Fintype m] [CommRing A] [IsDomain A] [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0) {v : m → A} (hv : M.mulVec v = 0) : v = 0

theorem LinearIndependent.sum_smul_of_nondegenerate {ι : Type u_1} {κ : Type u_2} {R : Type u_3} {M : Type u_4} [Fintype ι] [Finite κ] [CommRing R] [AddCommGroup M] [Module R M] {v : ι → M} (hv : LinearIndependent R v) {A : Matrix κ ι R} (hA : A.Nondegenerate) : LinearIndependent R fun (i : κ) => ∑ j : ι, A i j • v j