# 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.Nondegenerate {m : Type u_1} {R : Type u_2} [] [] (M : Matrix m m R) :

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

Equations
• M.Nondegenerate = ∀ (v : mR), (∀ (w : mR), Matrix.dotProduct v (M.mulVec w) = 0)v = 0
Instances For
theorem Matrix.Nondegenerate.eq_zero_of_ortho {m : Type u_1} {R : Type u_2} [] [] {M : Matrix m m R} (hM : M.Nondegenerate) {v : mR} (hv : ∀ (w : mR), Matrix.dotProduct 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} {R : Type u_2} [] [] {M : Matrix m m R} (hM : M.Nondegenerate) {v : mR} (hv : v 0) :
∃ (w : mR), Matrix.dotProduct 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_of_det_ne_zero {m : Type u_1} {A : Type u_3} [] [] [] [] {M : Matrix m m A} (hM : M.det 0) :
M.Nondegenerate

If M has a nonzero determinant, then M as a bilinear form on n → A is nondegenerate.

See also BilinForm.nondegenerateOfDetNeZero' and BilinForm.nondegenerateOfDetNeZero.

theorem Matrix.eq_zero_of_vecMul_eq_zero {m : Type u_1} {A : Type u_3} [] [] [] [] {M : Matrix m m A} (hM : M.det 0) {v : mA} (hv : = 0) :
v = 0
theorem Matrix.eq_zero_of_mulVec_eq_zero {m : Type u_1} {A : Type u_3} [] [] [] [] {M : Matrix m m A} (hM : M.det 0) {v : mA} (hv : M.mulVec v = 0) :
v = 0