mathlib documentation

linear_algebra.quadratic_form

Quadratic forms #

This file defines quadratic forms over a R-module M. A quadratic form is a map Q : M → R such that (to_fun_smul) Q (a • x) = a * a * Q x (polar_...) The map polar Q := λ x y, Q (x + y) - Q x - Q y is bilinear. They come with a scalar multiplication, (a • Q) x = Q (a • x) = a * a * Q x, and composition with linear maps f, Q.comp f x = Q (f x).

Main definitions #

quadratic_form.associated: associated bilinear form
quadratic_form.pos_def: positive definite quadratic forms
quadratic_form.anisotropic: anisotropic quadratic forms
quadratic_form.discr: discriminant of a quadratic form

Main statements #

quadratic_form.associated_left_inverse,
quadratic_form.associated_right_inverse: in a commutative ring where 2 has an inverse, there is a correspondence between quadratic forms and symmetric bilinear forms
bilin_form.exists_orthogonal_basis: There exists an orthogonal basis with respect to any nondegenerate, symmetric bilinear form B.

Notation #

In this file, the variable R is used when a ring structure is sufficient and R₁ is used when specifically a comm_ring is required. This allows us to keep [module R M] and [module R₁ M] assumptions in the variables without confusion between * from ring and * from comm_ring.

The variable S is used when R itself has a • action.

References #

Tags #

quadratic form, homogeneous polynomial, quadratic polynomial

def quadratic_form.polar {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] (f : M → R) (x y : M) :

R

Up to a factor 2, Q.polar is the associated bilinear form for a quadratic form Q.d

Source of this name: https://en.wikipedia.org/wiki/Quadratic_form#Generalization

Equations

quadratic_form.polar f x y = f (x + y) - f x - f y

theorem quadratic_form.polar_add {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] (f g : M → R) (x y : M) :

quadratic_form.polar (f + g) x y = quadratic_form.polar f x y + quadratic_form.polar g x y

theorem quadratic_form.polar_neg {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] (f : M → R) (x y : M) :

quadratic_form.polar (-f) x y = -quadratic_form.polar f x y

theorem quadratic_form.polar_smul {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [monoid S] [distrib_mul_action S R] (f : M → R) (s : S) (x y : M) :

quadratic_form.polar (s • f) x y = s • quadratic_form.polar f x y

theorem quadratic_form.polar_comm {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] (f : M → R) (x y : M) :

quadratic_form.polar f x y = quadratic_form.polar f y x

structure quadratic_form (R : Type u) (M : Type v) [ring R] [add_comm_group M] [module R M] :

Type (max u v)

to_fun : M → R
to_fun_smul : ∀ (a : R) (x : M), c.to_fun (a • x) = (a * a) * c.to_fun x
polar_add_left' : ∀ (x x' y : M), quadratic_form.polar c.to_fun (x + x') y = quadratic_form.polar c.to_fun x y + quadratic_form.polar c.to_fun x' y
polar_smul_left' : ∀ (a : R) (x y : M), quadratic_form.polar c.to_fun (a • x) y = a • quadratic_form.polar c.to_fun x y
polar_add_right' : ∀ (x y y' : M), quadratic_form.polar c.to_fun x (y + y') = quadratic_form.polar c.to_fun x y + quadratic_form.polar c.to_fun x y'
polar_smul_right' : ∀ (a : R) (x y : M), quadratic_form.polar c.to_fun x (a • y) = a • quadratic_form.polar c.to_fun x y

A quadratic form over a module.

@[protected, instance]

def quadratic_form.has_coe_to_fun {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] :

has_coe_to_fun (quadratic_form R M)

Equations

quadratic_form.has_coe_to_fun = {F := λ (x : quadratic_form R M), M → R, coe := quadratic_form.to_fun _inst_4}

@[simp]

theorem quadratic_form.to_fun_eq_apply {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} :

Q.to_fun = ⇑Q

The simp normal form for a quadratic form is coe_fn, not to_fun.

theorem quadratic_form.map_smul {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (a : R) (x : M) :

⇑Q (a • x) = (a * a) * ⇑Q x

theorem quadratic_form.map_add_self {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (x : M) :

⇑Q (x + x) = 4 * ⇑Q x

@[simp]

theorem quadratic_form.map_zero {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} :

⇑Q 0 = 0

@[simp]

theorem quadratic_form.map_neg {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (x : M) :

⇑Q (-x) = ⇑Q x

theorem quadratic_form.map_sub {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (x y : M) :

⇑Q (x - y) = ⇑Q (y - x)

@[simp]

theorem quadratic_form.polar_zero_left {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (y : M) :

quadratic_form.polar ⇑Q 0 y = 0

@[simp]

theorem quadratic_form.polar_add_left {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (x x' y : M) :

quadratic_form.polar ⇑Q (x + x') y = quadratic_form.polar ⇑Q x y + quadratic_form.polar ⇑Q x' y

@[simp]

theorem quadratic_form.polar_smul_left {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (a : R) (x y : M) :

quadratic_form.polar ⇑Q (a • x) y = a * quadratic_form.polar ⇑Q x y

@[simp]

theorem quadratic_form.polar_neg_left {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (x y : M) :

quadratic_form.polar ⇑Q (-x) y = -quadratic_form.polar ⇑Q x y

@[simp]

theorem quadratic_form.polar_sub_left {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (x x' y : M) :

quadratic_form.polar ⇑Q (x - x') y = quadratic_form.polar ⇑Q x y - quadratic_form.polar ⇑Q x' y

@[simp]

theorem quadratic_form.polar_zero_right {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (y : M) :

quadratic_form.polar ⇑Q y 0 = 0

@[simp]

theorem quadratic_form.polar_add_right {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (x y y' : M) :

quadratic_form.polar ⇑Q x (y + y') = quadratic_form.polar ⇑Q x y + quadratic_form.polar ⇑Q x y'

@[simp]

theorem quadratic_form.polar_smul_right {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (a : R) (x y : M) :

quadratic_form.polar ⇑Q x (a • y) = a * quadratic_form.polar ⇑Q x y

@[simp]

theorem quadratic_form.polar_neg_right {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (x y : M) :

quadratic_form.polar ⇑Q x (-y) = -quadratic_form.polar ⇑Q x y

@[simp]

theorem quadratic_form.polar_sub_right {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (x y y' : M) :

quadratic_form.polar ⇑Q x (y - y') = quadratic_form.polar ⇑Q x y - quadratic_form.polar ⇑Q x y'

@[simp]

theorem quadratic_form.polar_self {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} (x : M) :

quadratic_form.polar ⇑Q x x = 2 * ⇑Q x

@[simp]

theorem quadratic_form.polar_smul_left_of_tower {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} [comm_semiring S] [algebra S R] [module S M] [is_scalar_tower S R M] (a : S) (x y : M) :

quadratic_form.polar ⇑Q (a • x) y = a • quadratic_form.polar ⇑Q x y

@[simp]

theorem quadratic_form.polar_smul_right_of_tower {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q : quadratic_form R M} [comm_semiring S] [algebra S R] [module S M] [is_scalar_tower S R M] (a : S) (x y : M) :

quadratic_form.polar ⇑Q x (a • y) = a • quadratic_form.polar ⇑Q x y

@[ext]

theorem quadratic_form.ext {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {Q Q' : quadratic_form R M} (H : ∀ (x : M), ⇑Q x = ⇑Q' x) :

Q = Q'

@[protected, instance]

def quadratic_form.has_zero {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] :

has_zero (quadratic_form R M)

Equations

quadratic_form.has_zero = {zero := {to_fun := λ (x : M), 0, to_fun_smul := _, polar_add_left' := _, polar_smul_left' := _, polar_add_right' := _, polar_smul_right' := _}}

@[simp]

theorem quadratic_form.coe_fn_zero {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] :

⇑0 = 0

@[simp]

theorem quadratic_form.zero_apply {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (x : M) :

⇑0 x = 0

@[protected, instance]

def quadratic_form.inhabited {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] :

inhabited (quadratic_form R M)

Equations

quadratic_form.inhabited = {default := 0}

@[protected, instance]

def quadratic_form.has_add {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] :

has_add (quadratic_form R M)

Equations

quadratic_form.has_add = {add := λ (Q Q' : quadratic_form R M), {to_fun := ⇑Q + ⇑Q', to_fun_smul := _, polar_add_left' := _, polar_smul_left' := _, polar_add_right' := _, polar_smul_right' := _}}

@[simp]

theorem quadratic_form.coe_fn_add {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (Q Q' : quadratic_form R M) :

⇑(Q + Q') = ⇑Q + ⇑Q'

@[simp]

theorem quadratic_form.add_apply {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (Q Q' : quadratic_form R M) (x : M) :

⇑(Q + Q') x = ⇑Q x + ⇑Q' x

@[protected, instance]

def quadratic_form.has_neg {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] :

has_neg (quadratic_form R M)

Equations

quadratic_form.has_neg = {neg := λ (Q : quadratic_form R M), {to_fun := -⇑Q, to_fun_smul := _, polar_add_left' := _, polar_smul_left' := _, polar_add_right' := _, polar_smul_right' := _}}

@[simp]

theorem quadratic_form.coe_fn_neg {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (Q : quadratic_form R M) :

@[simp]

theorem quadratic_form.neg_apply {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (Q : quadratic_form R M) (x : M) :

(⇑-Q) x = -⇑Q x

@[protected, instance]

def quadratic_form.add_comm_group {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] :

add_comm_group (quadratic_form R M)

Equations

quadratic_form.add_comm_group = {add := has_add.add quadratic_form.has_add, add_assoc := _, zero := 0, zero_add := _, add_zero := _, nsmul := add_group.nsmul._default 0 has_add.add quadratic_form.add_comm_group._proof_4 quadratic_form.add_comm_group._proof_5, nsmul_zero' := _, nsmul_succ' := _, neg := has_neg.neg quadratic_form.has_neg, sub := add_group.sub._default has_add.add quadratic_form.add_comm_group._proof_8 0 quadratic_form.add_comm_group._proof_9 quadratic_form.add_comm_group._proof_10 (add_group.nsmul._default 0 has_add.add quadratic_form.add_comm_group._proof_4 quadratic_form.add_comm_group._proof_5) quadratic_form.add_comm_group._proof_11 quadratic_form.add_comm_group._proof_12 has_neg.neg, sub_eq_add_neg := _, gsmul := add_group.gsmul._default has_add.add quadratic_form.add_comm_group._proof_14 0 quadratic_form.add_comm_group._proof_15 quadratic_form.add_comm_group._proof_16 (add_group.nsmul._default 0 has_add.add quadratic_form.add_comm_group._proof_4 quadratic_form.add_comm_group._proof_5) quadratic_form.add_comm_group._proof_17 quadratic_form.add_comm_group._proof_18 has_neg.neg, gsmul_zero' := _, gsmul_succ' := _, gsmul_neg' := _, add_left_neg := _, add_comm := _}

@[simp]

theorem quadratic_form.coe_fn_sub {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (Q Q' : quadratic_form R M) :

⇑(Q - Q') = ⇑Q - ⇑Q'

@[simp]

theorem quadratic_form.sub_apply {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (Q Q' : quadratic_form R M) (x : M) :

⇑(Q - Q') x = ⇑Q x - ⇑Q' x

def quadratic_form.coe_fn_add_monoid_hom {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] :

quadratic_form R M →+ M → R

@coe_fn (quadratic_form R M) as an add_monoid_hom.

This API mirrors add_monoid_hom.coe_fn.

Equations

quadratic_form.coe_fn_add_monoid_hom = {to_fun := coe_fn quadratic_form.has_coe_to_fun, map_zero' := _, map_add' := _}

@[simp]

theorem quadratic_form.coe_fn_add_monoid_hom_apply {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (x : quadratic_form R M) (ᾰ : M) :

⇑quadratic_form.coe_fn_add_monoid_hom x ᾰ = ⇑x ᾰ

@[simp]

theorem quadratic_form.eval_add_monoid_hom_apply {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (m : M) (ᾰ : quadratic_form R M) :

⇑(quadratic_form.eval_add_monoid_hom m) ᾰ = ⇑ᾰ m

def quadratic_form.eval_add_monoid_hom {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (m : M) :

quadratic_form R M →+ R

Evaluation on a particular element of the module M is an additive map over quadratic forms.

Equations

quadratic_form.eval_add_monoid_hom m = (add_monoid_hom.apply (λ (m : M), R) m).comp quadratic_form.coe_fn_add_monoid_hom

@[simp]

theorem quadratic_form.coe_fn_sum {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {ι : Type u_1} (Q : ι → quadratic_form R M) (s : finset ι) :

⇑∑ (i : ι) in s, Q i = ∑ (i : ι) in s, ⇑(Q i)

@[simp]

theorem quadratic_form.sum_apply {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {ι : Type u_1} (Q : ι → quadratic_form R M) (s : finset ι) (x : M) :

(⇑∑ (i : ι) in s, Q i) x = ∑ (i : ι) in s, ⇑(Q i) x

@[protected, instance]

def quadratic_form.has_scalar {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [monoid S] [distrib_mul_action S R] [smul_comm_class S R R] :

has_scalar S (quadratic_form R M)

quadratic_form R M inherits the scalar action from any algebra over R.

When R is commutative, this provides an R-action via algebra.id.

Equations

quadratic_form.has_scalar = {smul := λ (a : S) (Q : quadratic_form R M), {to_fun := a • ⇑Q, to_fun_smul := _, polar_add_left' := _, polar_smul_left' := _, polar_add_right' := _, polar_smul_right' := _}}

@[simp]

theorem quadratic_form.coe_fn_smul {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [monoid S] [distrib_mul_action S R] [smul_comm_class S R R] (a : S) (Q : quadratic_form R M) :

⇑(a • Q) = a • ⇑Q

@[simp]

theorem quadratic_form.smul_apply {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [monoid S] [distrib_mul_action S R] [smul_comm_class S R R] (a : S) (Q : quadratic_form R M) (x : M) :

⇑(a • Q) x = a • ⇑Q x

@[protected, instance]

def quadratic_form.distrib_mul_action {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [monoid S] [distrib_mul_action S R] [smul_comm_class S R R] :

distrib_mul_action S (quadratic_form R M)

Equations

quadratic_form.distrib_mul_action = {to_mul_action := {to_has_scalar := quadratic_form.has_scalar _inst_8, one_smul := _, mul_smul := _}, smul_add := _, smul_zero := _}

@[protected, instance]

def quadratic_form.module {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [semiring S] [module S R] [smul_comm_class S R R] :

module S (quadratic_form R M)

Equations

quadratic_form.module = {to_distrib_mul_action := quadratic_form.distrib_mul_action _inst_8, add_smul := _, zero_smul := _}

def quadratic_form.comp {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {N : Type v} [add_comm_group N] [module R N] (Q : quadratic_form R N) (f : M →ₗ[R] N) :

quadratic_form R M

Compose the quadratic form with a linear function.

Equations

Q.comp f = {to_fun := λ (x : M), ⇑Q (⇑f x), to_fun_smul := _, polar_add_left' := _, polar_smul_left' := _, polar_add_right' := _, polar_smul_right' := _}

@[simp]

theorem quadratic_form.comp_apply {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {N : Type v} [add_comm_group N] [module R N] (Q : quadratic_form R N) (f : M →ₗ[R] N) (x : M) :

⇑(Q.comp f) x = ⇑Q (⇑f x)

def quadratic_form.mk_left {M : Type u_3} [add_comm_group M] {R₁ : Type u_4} [comm_ring R₁] [module R₁ M] (f : M → R₁) (to_fun_smul : ∀ (a : R₁) (x : M), f (a • x) = (a * a) * f x) (polar_add_left : ∀ (x x' y : M), quadratic_form.polar f (x + x') y = quadratic_form.polar f x y + quadratic_form.polar f x' y) (polar_smul_left : ∀ (a : R₁) (x y : M), quadratic_form.polar f (a • x) y = a * quadratic_form.polar f x y) :

quadratic_form R₁ M

Create a quadratic form in a commutative ring by proving only one side of the bilinearity.

Equations

quadratic_form.mk_left f to_fun_smul polar_add_left polar_smul_left = {to_fun := f, to_fun_smul := to_fun_smul, polar_add_left' := polar_add_left, polar_smul_left' := polar_smul_left, polar_add_right' := _, polar_smul_right' := _}

def quadratic_form.lin_mul_lin {M : Type u_3} [add_comm_group M] {R₁ : Type u_4} [comm_ring R₁] [module R₁ M] (f g : M →ₗ[R₁] R₁) :

quadratic_form R₁ M

The product of linear forms is a quadratic form.

Equations

quadratic_form.lin_mul_lin f g = quadratic_form.mk_left ((⇑f) * ⇑g) _ _ _

@[simp]

theorem quadratic_form.lin_mul_lin_apply {M : Type u_3} [add_comm_group M] {R₁ : Type u_4} [comm_ring R₁] [module R₁ M] (f g : M →ₗ[R₁] R₁) (x : M) :

⇑(quadratic_form.lin_mul_lin f g) x = (⇑f x) * ⇑g x

@[simp]

theorem quadratic_form.add_lin_mul_lin {M : Type u_3} [add_comm_group M] {R₁ : Type u_4} [comm_ring R₁] [module R₁ M] (f g h : M →ₗ[R₁] R₁) :

quadratic_form.lin_mul_lin (f + g) h = quadratic_form.lin_mul_lin f h + quadratic_form.lin_mul_lin g h

@[simp]

theorem quadratic_form.lin_mul_lin_add {M : Type u_3} [add_comm_group M] {R₁ : Type u_4} [comm_ring R₁] [module R₁ M] (f g h : M →ₗ[R₁] R₁) :

quadratic_form.lin_mul_lin f (g + h) = quadratic_form.lin_mul_lin f g + quadratic_form.lin_mul_lin f h

@[simp]

theorem quadratic_form.lin_mul_lin_comp {M : Type u_3} [add_comm_group M] {R₁ : Type u_4} [comm_ring R₁] [module R₁ M] {N : Type v} [add_comm_group N] [module R₁ N] (f g : M →ₗ[R₁] R₁) (h : N →ₗ[R₁] M) :

(quadratic_form.lin_mul_lin f g).comp h = quadratic_form.lin_mul_lin (f.comp h) (g.comp h)

def quadratic_form.proj {R₁ : Type u_4} [comm_ring R₁] {n : Type u_5} (i j : n) :

quadratic_form R₁ (n → R₁)

proj i j is the quadratic form mapping the vector x : n → R₁ to x i * x j

Equations

quadratic_form.proj i j = quadratic_form.lin_mul_lin (linear_map.proj i) (linear_map.proj j)

@[simp]

theorem quadratic_form.proj_apply {R₁ : Type u_4} [comm_ring R₁] {n : Type u_5} (i j : n) (x : n → R₁) :

⇑(quadratic_form.proj i j) x = (x i) * x j

Associated bilinear forms #

Over a commutative ring with an inverse of 2, the theory of quadratic forms is basically identical to that of symmetric bilinear forms. The map from quadratic forms to bilinear forms giving this identification is called the associated quadratic form.

theorem bilin_form.polar_to_quadratic_form {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {B : bilin_form R M} (x y : M) :

quadratic_form.polar (λ (x : M), ⇑B x x) x y = ⇑B x y + ⇑B y x

def bilin_form.to_quadratic_form {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (B : bilin_form R M) :

quadratic_form R M

A bilinear form gives a quadratic form by applying the argument twice.

Equations

B.to_quadratic_form = {to_fun := λ (x : M), ⇑B x x, to_fun_smul := _, polar_add_left' := _, polar_smul_left' := _, polar_add_right' := _, polar_smul_right' := _}

@[simp]

theorem bilin_form.to_quadratic_form_apply {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (B : bilin_form R M) (x : M) :

⇑(B.to_quadratic_form) x = ⇑B x x

def quadratic_form.associated_hom (S : Type u_1) {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [comm_semiring S] [algebra S R] [invertible 2] :

quadratic_form R M →ₗ[S] bilin_form R M

associated_hom is the map that sends a quadratic form on a module M over R to its associated symmetric bilinear form. As provided here, this has the structure of an S-linear map where S is a commutative subring of R.

Over a commutative ring, use associated, which gives an R-linear map. Over a general ring with no nontrivial distinguished commutative subring, use associated', which gives an additive homomorphism (or more precisely a ℤ-linear map.)

Equations

quadratic_form.associated_hom S = {to_fun := λ (Q : quadratic_form R M), {bilin := λ (x y : M), (⅟ 2) * quadratic_form.polar ⇑Q x y, bilin_add_left := _, bilin_smul_left := _, bilin_add_right := _, bilin_smul_right := _}, map_add' := _, map_smul' := _}

@[simp]

theorem quadratic_form.associated_apply {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [comm_semiring S] [algebra S R] [invertible 2] {Q : quadratic_form R M} (x y : M) :

⇑(⇑(quadratic_form.associated_hom S) Q) x y = (⅟ 2) * (⇑Q (x + y) - ⇑Q x - ⇑Q y)

theorem quadratic_form.associated_is_sym {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [comm_semiring S] [algebra S R] [invertible 2] {Q : quadratic_form R M} :

sym_bilin_form.is_sym (⇑(quadratic_form.associated_hom S) Q)

@[simp]

theorem quadratic_form.associated_comp {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [comm_semiring S] [algebra S R] [invertible 2] {Q : quadratic_form R M} {N : Type v} [add_comm_group N] [module R N] (f : N →ₗ[R] M) :

⇑(quadratic_form.associated_hom S) (Q.comp f) = (⇑(quadratic_form.associated_hom S) Q).comp f f

theorem quadratic_form.associated_to_quadratic_form {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [comm_semiring S] [algebra S R] [invertible 2] (B : bilin_form R M) (x y : M) :

⇑(⇑(quadratic_form.associated_hom S) B.to_quadratic_form) x y = (⅟ 2) * (⇑B x y + ⇑B y x)

theorem quadratic_form.associated_left_inverse {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [comm_semiring S] [algebra S R] [invertible 2] {B₁ : bilin_form R M} (h : sym_bilin_form.is_sym B₁) :

⇑(quadratic_form.associated_hom S) B₁.to_quadratic_form = B₁

theorem quadratic_form.associated_right_inverse {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [comm_semiring S] [algebra S R] [invertible 2] {Q : quadratic_form R M} :

(⇑(quadratic_form.associated_hom S) Q).to_quadratic_form = Q

theorem quadratic_form.associated_eq_self_apply {S : Type u_1} {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [comm_semiring S] [algebra S R] [invertible 2] {Q : quadratic_form R M} (x : M) :

⇑(⇑(quadratic_form.associated_hom S) Q) x x = ⇑Q x

def quadratic_form.associated' {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [invertible 2] :

quadratic_form R M →ₗ[ℤ ] bilin_form R M

associated' is the ℤ-linear map that sends a quadratic form on a module M over R to its associated symmetric bilinear form.

theorem quadratic_form.exists_quadratic_form_neq_zero {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [invertible 2] [nontrivial M] {Q : quadratic_form R M} (hB₁ : (⇑quadratic_form.associated' Q).nondegenerate) :

∃ (x : M), ⇑Q x ≠ 0

There exists a non-null vector with respect to any quadratic form Q whose associated bilinear form is non-degenerate, i.e. there exists x such that Q x ≠ 0.

def quadratic_form.associated {M : Type u_3} [add_comm_group M] {R₁ : Type u_4} [comm_ring R₁] [module R₁ M] [invertible 2] :

quadratic_form R₁ M →ₗ[R₁] bilin_form R₁ M

associated is the linear map that sends a quadratic form over a commutative ring to its associated symmetric bilinear form.

@[simp]

theorem quadratic_form.associated_lin_mul_lin {M : Type u_3} [add_comm_group M] {R₁ : Type u_4} [comm_ring R₁] [module R₁ M] [invertible 2] (f g : M →ₗ[R₁] R₁) :

⇑quadratic_form.associated (quadratic_form.lin_mul_lin f g) = ⅟ 2 • (bilin_form.lin_mul_lin f g + bilin_form.lin_mul_lin g f)

def quadratic_form.anisotropic {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (Q : quadratic_form R M) :

Prop

An anisotropic quadratic form is zero only on zero vectors.

Equations

Q.anisotropic = ∀ (x : M), ⇑Q x = 0 → x = 0

theorem quadratic_form.not_anisotropic_iff_exists {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (Q : quadratic_form R M) :

¬Q.anisotropic ↔ ∃ (x : M) (H : x ≠ 0), ⇑Q x = 0

theorem quadratic_form.nondegenerate_of_anisotropic {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [invertible 2] (Q : quadratic_form R M) (hB : Q.anisotropic) :

(⇑quadratic_form.associated' Q).nondegenerate

The associated bilinear form of an anisotropic quadratic form is nondegenerate.

def quadratic_form.pos_def {M : Type u_3} [add_comm_group M] {R₂ : Type u} [ordered_ring R₂] [module R₂ M] (Q₂ : quadratic_form R₂ M) :

Prop

A positive definite quadratic form is positive on nonzero vectors.

Equations

Q₂.pos_def = ∀ (x : M), x ≠ 0 → 0 < ⇑Q₂ x

theorem quadratic_form.pos_def.smul {M : Type u_3} [add_comm_group M] {R : Type u_1} [linear_ordered_comm_ring R] [module R M] {Q : quadratic_form R M} (h : Q.pos_def) {a : R} (a_pos : 0 < a) :

(a • Q).pos_def

theorem quadratic_form.pos_def.add {M : Type u_3} [add_comm_group M] {R₂ : Type u} [ordered_ring R₂] [module R₂ M] (Q Q' : quadratic_form R₂ M) (hQ : Q.pos_def) (hQ' : Q'.pos_def) :

(Q + Q').pos_def

theorem quadratic_form.lin_mul_lin_self_pos_def {M : Type u_3} [add_comm_group M] {R : Type u_1} [linear_ordered_comm_ring R] [module R M] (f : M →ₗ[R] R) (hf : f.ker = ⊥) :

(quadratic_form.lin_mul_lin f f).pos_def

Quadratic forms and matrices #

Connect quadratic forms and matrices, in order to explicitly compute with them. The convention is twos out, so there might be a factor 2⁻¹ in the entries of the matrix. The determinant of the matrix is the discriminant of the quadratic form.

def matrix.to_quadratic_form' {R₁ : Type u_4} [comm_ring R₁] {n : Type w} [fintype n] [decidable_eq n] (M : matrix n n R₁) :

quadratic_form R₁ (n → R₁)

M.to_quadratic_form is the map λ x, col x ⬝ M ⬝ row x as a quadratic form.

Equations

M.to_quadratic_form' = (⇑matrix.to_bilin' M).to_quadratic_form

def quadratic_form.to_matrix' {R₁ : Type u_4} [comm_ring R₁] {n : Type w} [fintype n] [decidable_eq n] [invertible 2] (Q : quadratic_form R₁ (n → R₁)) :

matrix n n R₁

A matrix representation of the quadratic form.

Equations

Q.to_matrix' = ⇑bilin_form.to_matrix' (⇑quadratic_form.associated Q)

theorem quadratic_form.to_matrix'_smul {R₁ : Type u_4} [comm_ring R₁] {n : Type w} [fintype n] [decidable_eq n] [invertible 2] (a : R₁) (Q : quadratic_form R₁ (n → R₁)) :

(a • Q).to_matrix' = a • Q.to_matrix'

@[simp]

theorem quadratic_form.to_matrix'_comp {R₁ : Type u_4} [comm_ring R₁] {n : Type w} [fintype n] [decidable_eq n] [invertible 2] {m : Type w} [decidable_eq m] [fintype m] (Q : quadratic_form R₁ (m → R₁)) (f : (n → R₁) →ₗ[R₁] m → R₁) :

(Q.comp f).to_matrix' = (⇑linear_map.to_matrix' f)ᵀ ⬝ Q.to_matrix' ⬝ ⇑linear_map.to_matrix' f

def quadratic_form.discr {R₁ : Type u_4} [comm_ring R₁] {n : Type w} [fintype n] [decidable_eq n] [invertible 2] (Q : quadratic_form R₁ (n → R₁)) :

R₁

The discriminant of a quadratic form generalizes the discriminant of a quadratic polynomial.

Equations

Q.discr = Q.to_matrix'.det

theorem quadratic_form.discr_smul {R₁ : Type u_4} [comm_ring R₁] {n : Type w} [fintype n] [decidable_eq n] [invertible 2] {Q : quadratic_form R₁ (n → R₁)} (a : R₁) :

(a • Q).discr = (a ^ fintype.card n) * Q.discr

theorem quadratic_form.discr_comp {R₁ : Type u_4} [comm_ring R₁] {n : Type w} [fintype n] [decidable_eq n] [invertible 2] {Q : quadratic_form R₁ (n → R₁)} (f : (n → R₁) →ₗ[R₁] n → R₁) :

(Q.comp f).discr = (((⇑linear_map.to_matrix' f).det) * (⇑linear_map.to_matrix' f).det) * Q.discr

@[nolint]

structure quadratic_form.isometry {R : Type u_2} [ring R] {M₁ : Type u_5} {M₂ : Type u_6} [add_comm_group M₁] [add_comm_group M₂] [module R M₁] [module R M₂] (Q₁ : quadratic_form R M₁) (Q₂ : quadratic_form R M₂) :

Type (max u_5 u_6)

to_linear_equiv : M₁ ≃ₗ[R] M₂
map_app' : ∀ (m : M₁), ⇑Q₂ (c.to_linear_equiv.to_fun m) = ⇑Q₁ m

An isometry between two quadratic spaces M₁, Q₁ and M₂, Q₂ over a ring R, is a linear equivalence between M₁ and M₂ that commutes with the quadratic forms.

def quadratic_form.equivalent {R : Type u_2} [ring R] {M₁ : Type u_5} {M₂ : Type u_6} [add_comm_group M₁] [add_comm_group M₂] [module R M₁] [module R M₂] (Q₁ : quadratic_form R M₁) (Q₂ : quadratic_form R M₂) :

Prop

Two quadratic forms over a ring R are equivalent if there exists an isometry between them: a linear equivalence that transforms one quadratic form into the other.

Equations

Q₁.equivalent Q₂ = nonempty (Q₁.isometry Q₂)

@[protected, instance]

def quadratic_form.isometry.linear_equiv.has_coe {R : Type u_2} [ring R] {M₁ : Type u_5} {M₂ : Type u_6} [add_comm_group M₁] [add_comm_group M₂] [module R M₁] [module R M₂] {Q₁ : quadratic_form R M₁} {Q₂ : quadratic_form R M₂} :

has_coe (Q₁.isometry Q₂) (M₁ ≃ₗ[R] M₂)

Equations

quadratic_form.isometry.linear_equiv.has_coe = {coe := quadratic_form.isometry.to_linear_equiv Q₂}

@[protected, instance]

def quadratic_form.isometry.has_coe_to_fun {R : Type u_2} [ring R] {M₁ : Type u_5} {M₂ : Type u_6} [add_comm_group M₁] [add_comm_group M₂] [module R M₁] [module R M₂] {Q₁ : quadratic_form R M₁} {Q₂ : quadratic_form R M₂} :

has_coe_to_fun (Q₁.isometry Q₂)

Equations

quadratic_form.isometry.has_coe_to_fun = {F := λ (_x : Q₁.isometry Q₂), M₁ → M₂, coe := λ (f : Q₁.isometry Q₂), ⇑↑f}

@[simp]

theorem quadratic_form.isometry.map_app {R : Type u_2} [ring R] {M₁ : Type u_5} {M₂ : Type u_6} [add_comm_group M₁] [add_comm_group M₂] [module R M₁] [module R M₂] {Q₁ : quadratic_form R M₁} {Q₂ : quadratic_form R M₂} (f : Q₁.isometry Q₂) (m : M₁) :

⇑Q₂ (⇑f m) = ⇑Q₁ m

def quadratic_form.isometry.refl {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (Q : quadratic_form R M) :

The identity isometry from a quadratic form to itself.

Equations

quadratic_form.isometry.refl Q = {to_linear_equiv := {to_fun := (linear_equiv.refl R M).to_fun, map_add' := _, map_smul' := _, inv_fun := (linear_equiv.refl R M).inv_fun, left_inv := _, right_inv := _}, map_app' := _}

def quadratic_form.isometry.symm {R : Type u_2} [ring R] {M₁ : Type u_5} {M₂ : Type u_6} [add_comm_group M₁] [add_comm_group M₂] [module R M₁] [module R M₂] {Q₁ : quadratic_form R M₁} {Q₂ : quadratic_form R M₂} (f : Q₁.isometry Q₂) :

Q₂.isometry Q₁

The inverse isometry of an isometry between two quadratic forms.

Equations

f.symm = {to_linear_equiv := {to_fun := ↑f.symm.to_fun, map_add' := _, map_smul' := _, inv_fun := ↑f.symm.inv_fun, left_inv := _, right_inv := _}, map_app' := _}

def quadratic_form.isometry.trans {R : Type u_2} [ring R] {M₁ : Type u_5} {M₂ : Type u_6} {M₃ : Type u_7} [add_comm_group M₁] [add_comm_group M₂] [add_comm_group M₃] [module R M₁] [module R M₂] [module R M₃] {Q₁ : quadratic_form R M₁} {Q₂ : quadratic_form R M₂} {Q₃ : quadratic_form R M₃} (f : Q₁.isometry Q₂) (g : Q₂.isometry Q₃) :

Q₁.isometry Q₃

The composition of two isometries between quadratic forms.

Equations

f.trans g = {to_linear_equiv := {to_fun := (↑f.trans ↑g).to_fun, map_add' := _, map_smul' := _, inv_fun := (↑f.trans ↑g).inv_fun, left_inv := _, right_inv := _}, map_app' := _}

theorem quadratic_form.equivalent.refl {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] (Q : quadratic_form R M) :

theorem quadratic_form.equivalent.symm {R : Type u_2} [ring R] {M₁ : Type u_5} {M₂ : Type u_6} [add_comm_group M₁] [add_comm_group M₂] [module R M₁] [module R M₂] {Q₁ : quadratic_form R M₁} {Q₂ : quadratic_form R M₂} (h : Q₁.equivalent Q₂) :

Q₂.equivalent Q₁

theorem quadratic_form.equivalent.trans {R : Type u_2} [ring R] {M₁ : Type u_5} {M₂ : Type u_6} {M₃ : Type u_7} [add_comm_group M₁] [add_comm_group M₂] [add_comm_group M₃] [module R M₁] [module R M₂] [module R M₃] {Q₁ : quadratic_form R M₁} {Q₂ : quadratic_form R M₂} {Q₃ : quadratic_form R M₃} (h : Q₁.equivalent Q₂) (h' : Q₂.equivalent Q₃) :

Q₁.equivalent Q₃

theorem bilin_form.nondegenerate_of_anisotropic {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] {B : bilin_form R M} (hB : B.to_quadratic_form.anisotropic) :

B.nondegenerate

A bilinear form is nondegenerate if the quadratic form it is associated with is anisotropic.

theorem bilin_form.exists_bilin_form_self_neq_zero {R : Type u_2} {M : Type u_3} [add_comm_group M] [ring R] [module R M] [htwo : invertible 2] [nontrivial M] {B : bilin_form R M} (hB₁ : B.nondegenerate) (hB₂ : sym_bilin_form.is_sym B) :

∃ (x : M), ¬B.is_ortho x x

There exists a non-null vector with respect to any symmetric, nondegenerate bilinear form B on a nontrivial module M over a ring R with invertible 2, i.e. there exists some x : M such that B x x ≠ 0.

theorem bilin_form.exists_orthogonal_basis' {V : Type u} {K : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] [hK : invertible 2] {B : bilin_form K V} (hB₁ : B.nondegenerate) (hB₂ : sym_bilin_form.is_sym B) :

∃ (v : basis (fin (finite_dimensional.finrank K V)) K V), B.is_Ortho ⇑v ∧ ∀ (i : fin (finite_dimensional.finrank K V)), ⇑B (⇑v i) (⇑v i) ≠ 0

theorem bilin_form.exists_orthogonal_basis {V : Type u} {K : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] [hK : invertible 2] {B : bilin_form K V} (hB₁ : B.nondegenerate) (hB₂ : sym_bilin_form.is_sym B) :

∃ (v : basis (fin (finite_dimensional.finrank K V)) K V), B.is_Ortho ⇑v

Given a nondegenerate symmetric bilinear form B on some vector space V over the field K with invertible 2, there exists an orthogonal basis with respect to B.