Quadratic forms

This file defines quadratic forms over a R-module M. A quadratic form on a ring R is a map Q : M → R such that:

• QuadraticForm.map_smul: Q (a • x) = a * a * Q x
• QuadraticForm.polar_add_left, QuadraticForm.polar_add_right, QuadraticForm.polar_smul_left, QuadraticForm.polar_smul_right: the map QuadraticForm.polar Q := fun x y ↦ Q (x + y) - Q x - Q y is bilinear.

This notion generalizes to semirings using the approach in [izhakian2016][] which requires that there be a (possibly non-unique) companion bilinear form B such that ∀ x y, Q (x + y) = Q x + Q y + B x y. Over a ring, this B is precisely QuadraticForm.polar Q.

To build a QuadraticForm from the polar axioms, use QuadraticForm.ofPolar.

Quadratic forms 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

• QuadraticForm.ofPolar: a more familiar constructor that works on rings
• QuadraticForm.associated: associated bilinear form
• QuadraticForm.PosDef: positive definite quadratic forms
• QuadraticForm.Anisotropic: anisotropic quadratic forms
• QuadraticForm.discr: discriminant of a quadratic form

Main statements

• QuadraticForm.associated_left_inverse,
• QuadraticForm.associated_rightInverse: in a commutative ring where 2 has an inverse, there is a correspondence between quadratic forms and symmetric bilinear forms
• BilinForm.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 CommRing 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 CommRing.

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

References

• https://en.wikipedia.org/wiki/Quadratic_form
• https://en.wikipedia.org/wiki/Discriminant#Quadratic_forms

Tags

quadratic form, homogeneous polynomial, quadratic polynomial

def QuadraticForm.polar {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] (f : M → R) (x : M) (y : M) : R

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

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

theorem QuadraticForm.polar_add {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] (f : M → R) (g : M → R) (x : M) (y : M) : QuadraticForm.polar (f + g) x y = QuadraticForm.polar f x y + QuadraticForm.polar g x y

theorem QuadraticForm.polar_neg {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] (f : M → R) (x : M) (y : M) : QuadraticForm.polar (-f) x y = -QuadraticForm.polar f x y

theorem QuadraticForm.polar_smul {S : Type u_1} {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Monoid S] [DistribMulAction S R] (f : M → R) (s : S) (x : M) (y : M) : QuadraticForm.polar (s • f) x y = s • QuadraticForm.polar f x y

theorem QuadraticForm.polar_comm {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] (f : M → R) (x : M) (y : M) : QuadraticForm.polar f x y = QuadraticForm.polar f y x

theorem QuadraticForm.polar_add_left_iff {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] {f : M → R} {x : M} {x' : M} {y : M} : QuadraticForm.polar f (x + x') y = QuadraticForm.polar f x y + QuadraticForm.polar f x' y ↔ f (x + x' + y) + (f x + f x' + f y) = f (x + x') + f (x' + y) + f (y + x)

Auxiliary lemma to express bilinearity of QuadraticForm.polar without subtraction.

theorem QuadraticForm.polar_comp {S : Type u_1} {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] {F : Type u_6} [Ring S] [AddMonoidHomClass F R S] (f : M → R) (g : F) (x : M) (y : M) : QuadraticForm.polar (↑g ∘ f) x y = ↑g (QuadraticForm.polar f x y)

structure QuadraticForm (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] : Type (max u v)

• toFun : M → R
• toFun_smul : ∀ (a : R) (x : M), QuadraticForm.toFun s (a • x) = a * a * QuadraticForm.toFun s x
• exists_companion' : ∃ B, ∀ (x y : M), QuadraticForm.toFun s (x + y) = QuadraticForm.toFun s x + QuadraticForm.toFun s y + BilinForm.bilin B x y

A quadratic form over a module.

For a more familiar constructor when R is a ring, see QuadraticForm.ofPolar.

instance QuadraticForm.funLike {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] : FunLike (QuadraticForm R M) M fun x => R

instance QuadraticForm.instCoeFunQuadraticFormForAll {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] : CoeFun (QuadraticForm R M) fun x => M → R

Helper instance for when there's too many metavariables to apply FunLike.hasCoeToFun directly.

@[simp]

theorem QuadraticForm.toFun_eq_coe {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) : Q.toFun = ↑Q

The simp normal form for a quadratic form is FunLike.coe, not toFun.

theorem QuadraticForm.ext {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {Q : QuadraticForm R M} {Q' : QuadraticForm R M} (H : ∀ (x : M), ↑Q x = ↑Q' x) : Q = Q'

theorem QuadraticForm.congr_fun {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {Q : QuadraticForm R M} {Q' : QuadraticForm R M} (h : Q = Q') (x : M) : ↑Q x = ↑Q' x

theorem QuadraticForm.ext_iff {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {Q : QuadraticForm R M} {Q' : QuadraticForm R M} : Q = Q' ↔ ∀ (x : M), ↑Q x = ↑Q' x

def QuadraticForm.copy {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) (Q' : M → R) (h : Q' = ↑Q) : QuadraticForm R M

Copy of a QuadraticForm with a new toFun equal to the old one. Useful to fix definitional equalities.

@[simp]

theorem QuadraticForm.coe_copy {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) (Q' : M → R) (h : Q' = ↑Q) : ↑(QuadraticForm.copy Q Q' h) = Q'

theorem QuadraticForm.copy_eq {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) (Q' : M → R) (h : Q' = ↑Q) : QuadraticForm.copy Q Q' h = Q

theorem QuadraticForm.map_smul {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) (a : R) (x : M) : ↑Q (a • x) = a * a * ↑Q x

theorem QuadraticForm.exists_companion {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) : ∃ B, ∀ (x y : M), ↑Q (x + y) = ↑Q x + ↑Q y + BilinForm.bilin B x y

theorem QuadraticForm.map_add_add_add_map {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) (x : M) (y : M) (z : M) : ↑Q (x + y + z) + (↑Q x + ↑Q y + ↑Q z) = ↑Q (x + y) + ↑Q (y + z) + ↑Q (z + x)

theorem QuadraticForm.map_add_self {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) (x : M) : ↑Q (x + x) = 4 * ↑Q x

theorem QuadraticForm.map_zero {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) : ↑Q 0 = 0

instance QuadraticForm.zeroHomClass {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] : ZeroHomClass (QuadraticForm R M) M R

theorem QuadraticForm.map_smul_of_tower {S : Type u_1} {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) [CommSemiring S] [Algebra S R] [Module S M] [IsScalarTower S R M] (a : S) (x : M) : ↑Q (a • x) = (a * a) • ↑Q x

@[simp]

theorem QuadraticForm.map_neg {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (x : M) : ↑Q (-x) = ↑Q x

theorem QuadraticForm.map_sub {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (x : M) (y : M) : ↑Q (x - y) = ↑Q (y - x)

@[simp]

theorem QuadraticForm.polar_zero_left {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (y : M) : QuadraticForm.polar (↑Q) 0 y = 0

@[simp]

theorem QuadraticForm.polar_add_left {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (x : M) (x' : M) (y : M) : QuadraticForm.polar (↑Q) (x + x') y = QuadraticForm.polar (↑Q) x y + QuadraticForm.polar (↑Q) x' y

@[simp]

theorem QuadraticForm.polar_smul_left {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (a : R) (x : M) (y : M) : QuadraticForm.polar (↑Q) (a • x) y = a * QuadraticForm.polar (↑Q) x y

@[simp]

theorem QuadraticForm.polar_neg_left {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (x : M) (y : M) : QuadraticForm.polar (↑Q) (-x) y = -QuadraticForm.polar (↑Q) x y

@[simp]

theorem QuadraticForm.polar_sub_left {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (x : M) (x' : M) (y : M) : QuadraticForm.polar (↑Q) (x - x') y = QuadraticForm.polar (↑Q) x y - QuadraticForm.polar (↑Q) x' y

@[simp]

theorem QuadraticForm.polar_zero_right {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (y : M) : QuadraticForm.polar (↑Q) y 0 = 0

@[simp]

theorem QuadraticForm.polar_add_right {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (x : M) (y : M) (y' : M) : QuadraticForm.polar (↑Q) x (y + y') = QuadraticForm.polar (↑Q) x y + QuadraticForm.polar (↑Q) x y'

@[simp]

theorem QuadraticForm.polar_smul_right {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (a : R) (x : M) (y : M) : QuadraticForm.polar (↑Q) x (a • y) = a * QuadraticForm.polar (↑Q) x y

@[simp]

theorem QuadraticForm.polar_neg_right {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (x : M) (y : M) : QuadraticForm.polar (↑Q) x (-y) = -QuadraticForm.polar (↑Q) x y

@[simp]

theorem QuadraticForm.polar_sub_right {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (x : M) (y : M) (y' : M) : QuadraticForm.polar (↑Q) x (y - y') = QuadraticForm.polar (↑Q) x y - QuadraticForm.polar (↑Q) x y'

@[simp]

theorem QuadraticForm.polar_self {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (x : M) : QuadraticForm.polar (↑Q) x x = 2 * ↑Q x

@[simp]

theorem QuadraticForm.polarBilin_apply {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (x : M) (y : M) : BilinForm.bilin (QuadraticForm.polarBilin Q) x y = QuadraticForm.polar (↑Q) x y

def QuadraticForm.polarBilin {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) : BilinForm R M

QuadraticForm.polar as a bilinear form

@[simp]

theorem QuadraticForm.polar_smul_left_of_tower {S : Type u_1} {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) [CommSemiring S] [Algebra S R] [Module S M] [IsScalarTower S R M] (a : S) (x : M) (y : M) : QuadraticForm.polar (↑Q) (a • x) y = a • QuadraticForm.polar (↑Q) x y

@[simp]

theorem QuadraticForm.polar_smul_right_of_tower {S : Type u_1} {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) [CommSemiring S] [Algebra S R] [Module S M] [IsScalarTower S R M] (a : S) (x : M) (y : M) : QuadraticForm.polar (↑Q) x (a • y) = a • QuadraticForm.polar (↑Q) x y

@[simp]

theorem QuadraticForm.ofPolar_apply {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (toFun : M → R) (toFun_smul : ∀ (a : R) (x : M), toFun (a • x) = a * a * toFun x) (polar_add_left : ∀ (x x' y : M), QuadraticForm.polar toFun (x + x') y = QuadraticForm.polar toFun x y + QuadraticForm.polar toFun x' y) (polar_smul_left : ∀ (a : R) (x y : M), QuadraticForm.polar toFun (a • x) y = a • QuadraticForm.polar toFun x y) : ∀ (a : M), ↑(QuadraticForm.ofPolar toFun toFun_smul polar_add_left polar_smul_left) a = toFun a

def QuadraticForm.ofPolar {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (toFun : M → R) (toFun_smul : ∀ (a : R) (x : M), toFun (a • x) = a * a * toFun x) (polar_add_left : ∀ (x x' y : M), QuadraticForm.polar toFun (x + x') y = QuadraticForm.polar toFun x y + QuadraticForm.polar toFun x' y) (polar_smul_left : ∀ (a : R) (x y : M), QuadraticForm.polar toFun (a • x) y = a • QuadraticForm.polar toFun x y) : QuadraticForm R M

An alternative constructor to QuadraticForm.mk, for rings where polar can be used.

theorem QuadraticForm.choose_exists_companion {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) : Exists.choose (_ : ∃ B, ∀ (x y : M), ↑Q (x + y) = ↑Q x + ↑Q y + BilinForm.bilin B x y) = QuadraticForm.polarBilin Q

In a ring the companion bilinear form is unique and equal to QuadraticForm.polar.

instance QuadraticForm.instSMulQuadraticForm {S : Type u_1} {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S R] [SMulCommClass S R R] : SMul S (QuadraticForm R M)

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

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

@[simp]

theorem QuadraticForm.coeFn_smul {S : Type u_1} {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S R] [SMulCommClass S R R] (a : S) (Q : QuadraticForm R M) : ↑(a • Q) = a • ↑Q

@[simp]

theorem QuadraticForm.smul_apply {S : Type u_1} {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S R] [SMulCommClass S R R] (a : S) (Q : QuadraticForm R M) (x : M) : ↑(a • Q) x = a • ↑Q x

instance QuadraticForm.instSMulCommClassQuadraticFormInstSMulQuadraticFormInstSMulQuadraticForm {S : Type u_1} {T : Type u_2} {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [Monoid T] [DistribMulAction S R] [DistribMulAction T R] [SMulCommClass S R R] [SMulCommClass T R R] [SMulCommClass S T R] : SMulCommClass S T (QuadraticForm R M)

instance QuadraticForm.instIsScalarTowerQuadraticFormInstSMulQuadraticFormInstSMulQuadraticForm {S : Type u_1} {T : Type u_2} {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [Monoid T] [DistribMulAction S R] [DistribMulAction T R] [SMulCommClass S R R] [SMulCommClass T R R] [SMul S T] [IsScalarTower S T R] : IsScalarTower S T (QuadraticForm R M)

instance QuadraticForm.instZeroQuadraticForm {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] : Zero (QuadraticForm R M)

@[simp]

theorem QuadraticForm.coeFn_zero {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] : ↑0 = 0

@[simp]

theorem QuadraticForm.zero_apply {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (x : M) : ↑0 x = 0

instance QuadraticForm.instInhabitedQuadraticForm {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] : Inhabited (QuadraticForm R M)

instance QuadraticForm.instAddQuadraticForm {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] : Add (QuadraticForm R M)

@[simp]

theorem QuadraticForm.coeFn_add {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) (Q' : QuadraticForm R M) : ↑(Q + Q') = ↑Q + ↑Q'

@[simp]

theorem QuadraticForm.add_apply {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) (Q' : QuadraticForm R M) (x : M) : ↑(Q + Q') x = ↑Q x + ↑Q' x

instance QuadraticForm.instAddCommMonoidQuadraticForm {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] : AddCommMonoid (QuadraticForm R M)

@[simp]

theorem QuadraticForm.coeFnAddMonoidHom_apply {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] : ∀ (a : QuadraticForm R M) (a_1 : M), ↑QuadraticForm.coeFnAddMonoidHom a a_1 = ↑a a_1

def QuadraticForm.coeFnAddMonoidHom {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] : QuadraticForm R M →+ M → R

@CoeFn (QuadraticForm R M) as an AddMonoidHom.

This API mirrors AddMonoidHom.coeFn.

@[simp]

theorem QuadraticForm.evalAddMonoidHom_apply {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (m : M) : ∀ (a : QuadraticForm R M), ↑(QuadraticForm.evalAddMonoidHom m) a = ↑a m

def QuadraticForm.evalAddMonoidHom {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (m : M) : QuadraticForm R M →+ R

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

@[simp]

theorem QuadraticForm.coeFn_sum {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_6} (Q : ι → QuadraticForm R M) (s : Finset ι) : ↑(Finset.sum s fun i => Q i) = Finset.sum s fun i => ↑(Q i)

@[simp]

theorem QuadraticForm.sum_apply {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_6} (Q : ι → QuadraticForm R M) (s : Finset ι) (x : M) : ↑(Finset.sum s fun i => Q i) x = Finset.sum s fun i => ↑(Q i) x

instance QuadraticForm.instDistribMulActionQuadraticFormToAddMonoidInstAddCommMonoidQuadraticForm {S : Type u_1} {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S R] [SMulCommClass S R R] : DistribMulAction S (QuadraticForm R M)

instance QuadraticForm.instModuleQuadraticFormInstAddCommMonoidQuadraticForm {S : Type u_1} {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Semiring S] [Module S R] [SMulCommClass S R R] : Module S (QuadraticForm R M)

instance QuadraticForm.instNegQuadraticFormToSemiringToAddCommMonoid {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] : Neg (QuadraticForm R M)

@[simp]

theorem QuadraticForm.coeFn_neg {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) : ↑(-Q) = -↑Q

@[simp]

theorem QuadraticForm.neg_apply {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (x : M) : ↑(-Q) x = -↑Q x

instance QuadraticForm.instSubQuadraticFormToSemiringToAddCommMonoid {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] : Sub (QuadraticForm R M)

@[simp]

theorem QuadraticForm.coeFn_sub {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (Q' : QuadraticForm R M) : ↑(Q - Q') = ↑Q - ↑Q'

@[simp]

theorem QuadraticForm.sub_apply {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (Q' : QuadraticForm R M) (x : M) : ↑(Q - Q') x = ↑Q x - ↑Q' x

instance QuadraticForm.instAddCommGroupQuadraticFormToSemiringToAddCommMonoid {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] : AddCommGroup (QuadraticForm R M)

def QuadraticForm.comp {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {N : Type v} [AddCommMonoid N] [Module R N] (Q : QuadraticForm R N) (f : M →ₗ[R] N) : QuadraticForm R M

Compose the quadratic form with a linear function.

@[simp]

theorem QuadraticForm.comp_apply {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {N : Type v} [AddCommMonoid N] [Module R N] (Q : QuadraticForm R N) (f : M →ₗ[R] N) (x : M) : ↑(QuadraticForm.comp Q f) x = ↑Q (↑f x)

@[simp]

theorem LinearMap.compQuadraticForm_apply {S : Type u_1} {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [CommSemiring S] [Algebra S R] [Module S M] [IsScalarTower S R M] (f : R →ₗ[S] S) (Q : QuadraticForm R M) (x : M) : ↑(LinearMap.compQuadraticForm f Q) x = ↑f (↑Q x)

def LinearMap.compQuadraticForm {S : Type u_1} {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [CommSemiring S] [Algebra S R] [Module S M] [IsScalarTower S R M] (f : R →ₗ[S] S) (Q : QuadraticForm R M) : QuadraticForm S M

Compose a quadratic form with a linear function on the left.

def QuadraticForm.linMulLin {R : Type u_3} {M : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] (f : M →ₗ[R] R) (g : M →ₗ[R] R) : QuadraticForm R M

The product of linear forms is a quadratic form.

@[simp]

theorem QuadraticForm.linMulLin_apply {R : Type u_3} {M : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] (f : M →ₗ[R] R) (g : M →ₗ[R] R) (x : M) : ↑(QuadraticForm.linMulLin f g) x = ↑f x * ↑g x

@[simp]

theorem QuadraticForm.add_linMulLin {R : Type u_3} {M : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] (f : M →ₗ[R] R) (g : M →ₗ[R] R) (h : M →ₗ[R] R) : QuadraticForm.linMulLin (f + g) h = QuadraticForm.linMulLin f h + QuadraticForm.linMulLin g h

@[simp]

theorem QuadraticForm.linMulLin_add {R : Type u_3} {M : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] (f : M →ₗ[R] R) (g : M →ₗ[R] R) (h : M →ₗ[R] R) : QuadraticForm.linMulLin f (g + h) = QuadraticForm.linMulLin f g + QuadraticForm.linMulLin f h

@[simp]

theorem QuadraticForm.linMulLin_comp {R : Type u_3} {M : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] {N : Type v} [AddCommMonoid N] [Module R N] (f : M →ₗ[R] R) (g : M →ₗ[R] R) (h : N →ₗ[R] M) : QuadraticForm.comp (QuadraticForm.linMulLin f g) h = QuadraticForm.linMulLin (LinearMap.comp f h) (LinearMap.comp g h)

@[simp]

theorem QuadraticForm.sq_apply {R : Type u_3} [CommSemiring R] (a : R) : ↑QuadraticForm.sq a = a * a

def QuadraticForm.sq {R : Type u_3} [CommSemiring R] : QuadraticForm R R

sq is the quadratic form mapping the vector x : R₁ to x * x

def QuadraticForm.proj {R : Type u_3} [CommSemiring R] {n : Type u_6} (i : n) (j : n) : QuadraticForm R (n → R)

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

@[simp]

theorem QuadraticForm.proj_apply {R : Type u_3} [CommSemiring R] {n : Type u_6} (i : n) (j : n) (x : n → R) : ↑(QuadraticForm.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.

def BilinForm.toQuadraticForm {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (B : BilinForm R M) : QuadraticForm R M

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

@[simp]

theorem BilinForm.toQuadraticForm_apply {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (B : BilinForm R M) (x : M) : ↑(BilinForm.toQuadraticForm B) x = BilinForm.bilin B x x

@[simp]

theorem BilinForm.toQuadraticForm_zero (R : Type u_3) (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] : BilinForm.toQuadraticForm 0 = 0

@[simp]

theorem BilinForm.toQuadraticForm_add {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (B₁ : BilinForm R M) (B₂ : BilinForm R M) : BilinForm.toQuadraticForm (B₁ + B₂) = BilinForm.toQuadraticForm B₁ + BilinForm.toQuadraticForm B₂

@[simp]

theorem BilinForm.toQuadraticForm_smul {S : Type u_1} {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S R] [SMulCommClass S R R] (a : S) (B : BilinForm R M) : BilinForm.toQuadraticForm (a • B) = a • BilinForm.toQuadraticForm B

@[simp]

theorem BilinForm.toQuadraticFormAddMonoidHom_apply (R : Type u_3) (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] (B : BilinForm R M) : ↑(BilinForm.toQuadraticFormAddMonoidHom R M) B = BilinForm.toQuadraticForm B

def BilinForm.toQuadraticFormAddMonoidHom (R : Type u_3) (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] : BilinForm R M →+ QuadraticForm R M

BilinForm.toQuadraticForm as an additive homomorphism

@[simp]

theorem BilinForm.toQuadraticFormLinearMap_apply_apply (S : Type u_1) (R : Type u_3) (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] [Semiring S] [Module S R] [SMulCommClass S R R] (B : BilinForm R M) (x : M) : ↑(↑(BilinForm.toQuadraticFormLinearMap S R M) B) x = BilinForm.bilin B x x

def BilinForm.toQuadraticFormLinearMap (S : Type u_1) (R : Type u_3) (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] [Semiring S] [Module S R] [SMulCommClass S R R] : BilinForm R M →ₗ[S] QuadraticForm R M

BilinForm.toQuadraticForm as a linear map

@[simp]

theorem BilinForm.toQuadraticForm_list_sum {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (B : List (BilinForm R M)) : BilinForm.toQuadraticForm (List.sum B) = List.sum (List.map BilinForm.toQuadraticForm B)

@[simp]

theorem BilinForm.toQuadraticForm_multiset_sum {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (B : Multiset (BilinForm R M)) : BilinForm.toQuadraticForm (Multiset.sum B) = Multiset.sum (Multiset.map BilinForm.toQuadraticForm B)

@[simp]

theorem BilinForm.toQuadraticForm_sum {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_6} (s : Finset ι) (B : ι → BilinForm R M) : BilinForm.toQuadraticForm (Finset.sum s fun i => B i) = Finset.sum s fun i => BilinForm.toQuadraticForm (B i)

@[simp]

theorem BilinForm.toQuadraticForm_eq_zero {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {B : BilinForm R M} : BilinForm.toQuadraticForm B = 0 ↔ BilinForm.IsAlt B

@[simp]

theorem BilinForm.toQuadraticForm_neg {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (B : BilinForm R M) : BilinForm.toQuadraticForm (-B) = -BilinForm.toQuadraticForm B

@[simp]

theorem BilinForm.toQuadraticForm_sub {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (B₁ : BilinForm R M) (B₂ : BilinForm R M) : BilinForm.toQuadraticForm (B₁ - B₂) = BilinForm.toQuadraticForm B₁ - BilinForm.toQuadraticForm B₂

theorem BilinForm.polar_toQuadraticForm {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] {B : BilinForm R M} (x : M) (y : M) : QuadraticForm.polar (↑(BilinForm.toQuadraticForm B)) x y = BilinForm.bilin B x y + BilinForm.bilin B y x

theorem BilinForm.polarBilin_toQuadraticForm {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] {B : BilinForm R M} : QuadraticForm.polarBilin (BilinForm.toQuadraticForm B) = B + ↑BilinForm.flip' B

@[simp]

theorem QuadraticForm.toQuadraticForm_polarBilin {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) : BilinForm.toQuadraticForm (QuadraticForm.polarBilin Q) = 2 • Q

theorem QuadraticForm.polarBilin_injective {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] (h : IsUnit 2) : Function.Injective QuadraticForm.polarBilin

theorem QuadraticForm.polarBilin_comp {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] {N : Type v} [AddCommGroup N] [Module R N] (Q : QuadraticForm R N) (f : M →ₗ[R] N) : QuadraticForm.polarBilin (QuadraticForm.comp Q f) = BilinForm.comp (QuadraticForm.polarBilin Q) f f

theorem LinearMap.compQuadraticForm_polar {S : Type u_1} {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [CommRing S] [Algebra S R] [Module S M] [IsScalarTower S R M] (f : R →ₗ[S] S) (Q : QuadraticForm R M) (x : M) (y : M) : QuadraticForm.polar (↑(LinearMap.compQuadraticForm f Q)) x y = ↑f (QuadraticForm.polar (↑Q) x y)

theorem LinearMap.compQuadraticForm_polarBilin {S : Type u_1} {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [CommRing S] [Algebra S R] [Module S M] [IsScalarTower S R M] (f : R →ₗ[S] S) (Q : QuadraticForm R M) : QuadraticForm.polarBilin (LinearMap.compQuadraticForm f Q) = LinearMap.compBilinForm f (QuadraticForm.polarBilin Q)

def QuadraticForm.associatedHom (S : Type u_1) {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] : QuadraticForm R M →ₗ[S] BilinForm R M

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

@[simp]

theorem QuadraticForm.associated_apply (S : Type u_1) {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] (Q : QuadraticForm R M) (x : M) (y : M) : BilinForm.bilin (↑(QuadraticForm.associatedHom S) Q) x y = ⅟2 * (↑Q (x + y) - ↑Q x - ↑Q y)

@[simp]

theorem QuadraticForm.two_nsmul_associated (S : Type u_1) {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] (Q : QuadraticForm R M) : 2 • ↑(QuadraticForm.associatedHom S) Q = QuadraticForm.polarBilin Q

theorem QuadraticForm.associated_isSymm (S : Type u_1) {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] (Q : QuadraticForm R M) : BilinForm.IsSymm (↑(QuadraticForm.associatedHom S) Q)

@[simp]

theorem QuadraticForm.associated_comp (S : Type u_1) {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] (Q : QuadraticForm R M) {N : Type v} [AddCommGroup N] [Module R N] (f : N →ₗ[R] M) : ↑(QuadraticForm.associatedHom S) (QuadraticForm.comp Q f) = BilinForm.comp (↑(QuadraticForm.associatedHom S) Q) f f

theorem QuadraticForm.associated_toQuadraticForm (S : Type u_1) {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] (B : BilinForm R M) (x : M) (y : M) : BilinForm.bilin (↑(QuadraticForm.associatedHom S) (BilinForm.toQuadraticForm B)) x y = ⅟2 * (BilinForm.bilin B x y + BilinForm.bilin B y x)

theorem QuadraticForm.associated_left_inverse (S : Type u_1) {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] {B₁ : BilinForm R M} (h : BilinForm.IsSymm B₁) : ↑(QuadraticForm.associatedHom S) (BilinForm.toQuadraticForm B₁) = B₁

theorem QuadraticForm.associated_eq_self_apply (S : Type u_1) {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] (Q : QuadraticForm R M) (x : M) : BilinForm.bilin (↑(QuadraticForm.associatedHom S) Q) x x = ↑Q x

theorem QuadraticForm.toQuadraticForm_associated (S : Type u_1) {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] (Q : QuadraticForm R M) : BilinForm.toQuadraticForm (↑(QuadraticForm.associatedHom S) Q) = Q

theorem QuadraticForm.associated_rightInverse (S : Type u_1) {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] : Function.RightInverse (↑(QuadraticForm.associatedHom S)) BilinForm.toQuadraticForm

@[inline, reducible]

abbrev QuadraticForm.associated' {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [Invertible 2] : QuadraticForm R M →ₗ[ℤ] BilinForm 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.

instance QuadraticForm.canLift {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [Invertible 2] : CanLift (BilinForm R M) (QuadraticForm R M) (↑(QuadraticForm.associatedHom ℕ)) BilinForm.IsSymm

Symmetric bilinear forms can be lifted to quadratic forms

theorem QuadraticForm.exists_quadraticForm_ne_zero {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [Invertible 2] {Q : QuadraticForm R M} (hB₁ : ↑QuadraticForm.associated' Q ≠ 0) : ∃ x, ↑Q x ≠ 0

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

@[inline, reducible]

abbrev QuadraticForm.associated {R₁ : Type u_4} {M : Type u_5} [CommRing R₁] [AddCommGroup M] [Module R₁ M] [Invertible 2] : QuadraticForm R₁ M →ₗ[R₁] BilinForm R₁ M

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

theorem QuadraticForm.coe_associatedHom (S : Type u_1) {R₁ : Type u_4} {M : Type u_5} [CommSemiring S] [CommRing R₁] [AddCommGroup M] [Algebra S R₁] [Module R₁ M] [Invertible 2] : ↑(QuadraticForm.associatedHom S) = ↑QuadraticForm.associated

@[simp]

theorem QuadraticForm.associated_linMulLin {R₁ : Type u_4} {M : Type u_5} [CommRing R₁] [AddCommGroup M] [Module R₁ M] [Invertible 2] (f : M →ₗ[R₁] R₁) (g : M →ₗ[R₁] R₁) : ↑QuadraticForm.associated (QuadraticForm.linMulLin f g) = ⅟2 • (BilinForm.linMulLin f g + BilinForm.linMulLin g f)

@[simp]

theorem QuadraticForm.associated_sq {R₁ : Type u_4} [CommRing R₁] [Invertible 2] : ↑QuadraticForm.associated QuadraticForm.sq = ↑LinearMap.toBilin (LinearMap.mul R₁ R₁)

def QuadraticForm.Anisotropic {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) : Prop

An anisotropic quadratic form is zero only on zero vectors.

theorem QuadraticForm.not_anisotropic_iff_exists {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (Q : QuadraticForm R M) : ¬QuadraticForm.Anisotropic Q ↔ ∃ x, x ≠ 0 ∧ ↑Q x = 0

theorem QuadraticForm.Anisotropic.eq_zero_iff {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {Q : QuadraticForm R M} (h : QuadraticForm.Anisotropic Q) {x : M} : ↑Q x = 0 ↔ x = 0

theorem QuadraticForm.nondegenerate_of_anisotropic {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [Invertible 2] (Q : QuadraticForm R M) (hB : QuadraticForm.Anisotropic Q) : BilinForm.Nondegenerate (↑QuadraticForm.associated' Q)

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

def QuadraticForm.PosDef {M : Type u_5} {R₂ : Type u} [OrderedRing R₂] [AddCommMonoid M] [Module R₂ M] (Q₂ : QuadraticForm R₂ M) : Prop

A positive definite quadratic form is positive on nonzero vectors.

theorem QuadraticForm.PosDef.smul {M : Type u_5} [AddCommMonoid M] {R : Type u_6} [LinearOrderedCommRing R] [Module R M] {Q : QuadraticForm R M} (h : QuadraticForm.PosDef Q) {a : R} (a_pos : 0 < a) : QuadraticForm.PosDef (a • Q)

theorem QuadraticForm.PosDef.nonneg {M : Type u_5} {R₂ : Type u} [OrderedRing R₂] [AddCommMonoid M] [Module R₂ M] {Q : QuadraticForm R₂ M} (hQ : QuadraticForm.PosDef Q) (x : M) : 0 ≤ ↑Q x

theorem QuadraticForm.PosDef.anisotropic {M : Type u_5} {R₂ : Type u} [OrderedRing R₂] [AddCommMonoid M] [Module R₂ M] {Q : QuadraticForm R₂ M} (hQ : QuadraticForm.PosDef Q) : QuadraticForm.Anisotropic Q

theorem QuadraticForm.posDef_of_nonneg {M : Type u_5} {R₂ : Type u} [OrderedRing R₂] [AddCommMonoid M] [Module R₂ M] {Q : QuadraticForm R₂ M} (h : ∀ (x : M), 0 ≤ ↑Q x) (h0 : QuadraticForm.Anisotropic Q) : QuadraticForm.PosDef Q

theorem QuadraticForm.posDef_iff_nonneg {M : Type u_5} {R₂ : Type u} [OrderedRing R₂] [AddCommMonoid M] [Module R₂ M] {Q : QuadraticForm R₂ M} : QuadraticForm.PosDef Q ↔ (∀ (x : M), 0 ≤ ↑Q x) ∧ QuadraticForm.Anisotropic Q

theorem QuadraticForm.PosDef.add {M : Type u_5} {R₂ : Type u} [OrderedRing R₂] [AddCommMonoid M] [Module R₂ M] (Q : QuadraticForm R₂ M) (Q' : QuadraticForm R₂ M) (hQ : QuadraticForm.PosDef Q) (hQ' : QuadraticForm.PosDef Q') : QuadraticForm.PosDef (Q + Q')

theorem QuadraticForm.linMulLinSelfPosDef {M : Type u_5} [AddCommMonoid M] {R : Type u_7} [LinearOrderedCommRing R] [Module R M] (f : M →ₗ[R] R) (hf : LinearMap.ker f = ⊥) : QuadraticForm.PosDef (QuadraticForm.linMulLin f f)

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.toQuadraticForm' {R₁ : Type u_4} {n : Type w} [Fintype n] [DecidableEq n] [CommRing R₁] (M : Matrix n n R₁) : QuadraticForm R₁ (n → R₁)

M.toQuadraticForm' is the map fun x ↦ col x * M * row x as a quadratic form.

def QuadraticForm.toMatrix' {R₁ : Type u_4} {n : Type w} [Fintype n] [DecidableEq n] [CommRing R₁] [Invertible 2] (Q : QuadraticForm R₁ (n → R₁)) : Matrix n n R₁

A matrix representation of the quadratic form.

theorem QuadraticForm.toMatrix'_smul {R₁ : Type u_4} {n : Type w} [Fintype n] [DecidableEq n] [CommRing R₁] [Invertible 2] (a : R₁) (Q : QuadraticForm R₁ (n → R₁)) : QuadraticForm.toMatrix' (a • Q) = a • QuadraticForm.toMatrix' Q

theorem QuadraticForm.isSymm_toMatrix' {R₁ : Type u_4} {n : Type w} [Fintype n] [DecidableEq n] [CommRing R₁] [Invertible 2] (Q : QuadraticForm R₁ (n → R₁)) : Matrix.IsSymm (QuadraticForm.toMatrix' Q)

@[simp]

theorem QuadraticForm.toMatrix'_comp {R₁ : Type u_4} {n : Type w} [Fintype n] [CommRing R₁] [DecidableEq n] [Invertible 2] {m : Type w} [DecidableEq m] [Fintype m] (Q : QuadraticForm R₁ (m → R₁)) (f : (n → R₁) →ₗ[R₁] m → R₁) : QuadraticForm.toMatrix' (QuadraticForm.comp Q f) = Matrix.transpose (↑LinearMap.toMatrix' f) * QuadraticForm.toMatrix' Q * ↑LinearMap.toMatrix' f

def QuadraticForm.discr {R₁ : Type u_4} {n : Type w} [Fintype n] [CommRing R₁] [DecidableEq n] [Invertible 2] (Q : QuadraticForm R₁ (n → R₁)) : R₁

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

theorem QuadraticForm.discr_smul {R₁ : Type u_4} {n : Type w} [Fintype n] [CommRing R₁] [DecidableEq n] [Invertible 2] {Q : QuadraticForm R₁ (n → R₁)} (a : R₁) : QuadraticForm.discr (a • Q) = a ^ Fintype.card n * QuadraticForm.discr Q

theorem QuadraticForm.discr_comp {R₁ : Type u_4} {n : Type w} [Fintype n] [CommRing R₁] [DecidableEq n] [Invertible 2] {Q : QuadraticForm R₁ (n → R₁)} (f : (n → R₁) →ₗ[R₁] n → R₁) : QuadraticForm.discr (QuadraticForm.comp Q f) = Matrix.det (↑LinearMap.toMatrix' f) * Matrix.det (↑LinearMap.toMatrix' f) * QuadraticForm.discr Q

theorem BilinForm.nondegenerate_of_anisotropic {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {B : BilinForm R M} (hB : QuadraticForm.Anisotropic (BilinForm.toQuadraticForm B)) : BilinForm.Nondegenerate B

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

theorem BilinForm.exists_bilinForm_self_ne_zero {R : Type u_3} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] [htwo : Invertible 2] {B : BilinForm R M} (hB₁ : B ≠ 0) (hB₂ : BilinForm.IsSymm B) : ∃ x, ¬BilinForm.IsOrtho B x x

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

theorem BilinForm.exists_orthogonal_basis {V : Type u} {K : Type v} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] [hK : Invertible 2] {B : BilinForm K V} (hB₂ : BilinForm.IsSymm B) : ∃ v, BilinForm.iIsOrtho B ↑v

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

noncomputable def QuadraticForm.basisRepr {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_7} [Fintype ι] (Q : QuadraticForm R M) (v : Basis ι R M) : QuadraticForm R (ι → R)

Given a quadratic form Q and a basis, basisRepr is the basis representation of Q.

@[simp]

theorem QuadraticForm.basisRepr_apply {R : Type u_3} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_7} [Fintype ι] {v : Basis ι R M} (Q : QuadraticForm R M) (w : ι → R) : ↑(QuadraticForm.basisRepr Q v) w = ↑Q (Finset.sum Finset.univ fun i => w i • ↑v i)

def QuadraticForm.weightedSumSquares {S : Type u_1} (R₁ : Type u_4) [CommSemiring R₁] {ι : Type u_7} [Fintype ι] [Monoid S] [DistribMulAction S R₁] [SMulCommClass S R₁ R₁] (w : ι → S) : QuadraticForm R₁ (ι → R₁)

The weighted sum of squares with respect to some weight as a quadratic form.

The weights are applied using •; typically this definition is used either with S = R₁ or [Algebra S R₁], although this is stated more generally.

@[simp]

theorem QuadraticForm.weightedSumSquares_apply {S : Type u_1} {R₁ : Type u_4} [CommSemiring R₁] {ι : Type u_7} [Fintype ι] [Monoid S] [DistribMulAction S R₁] [SMulCommClass S R₁ R₁] (w : ι → S) (v : ι → R₁) : ↑(QuadraticForm.weightedSumSquares R₁ w) v = Finset.sum Finset.univ fun i => w i • (v i * v i)

theorem QuadraticForm.basisRepr_eq_of_iIsOrtho {ι : Type u_7} [Fintype ι] {R₁ : Type u_8} {M : Type u_9} [CommRing R₁] [AddCommGroup M] [Module R₁ M] [Invertible 2] (Q : QuadraticForm R₁ M) (v : Basis ι R₁ M) (hv₂ : BilinForm.iIsOrtho (↑QuadraticForm.associated Q) ↑v) : QuadraticForm.basisRepr Q v = QuadraticForm.weightedSumSquares R₁ fun i => ↑Q (↑v i)

On an orthogonal basis, the basis representation of Q is just a sum of squares.