Multilinear maps #
We define multilinear maps as maps from ∀ (i : ι), M₁ i to M₂ which are linear in each
coordinate. Here, M₁ i and M₂ are modules over a ring R, and ι is an arbitrary type
(although some statements will require it to be a fintype). This space, denoted by
MultilinearMap R M₁ M₂, inherits a module structure by pointwise addition and multiplication.
Main definitions #
MultilinearMap R M₁ M₂is the space of multilinear maps from∀ (i : ι), M₁ itoM₂.f.map_update_smulis the multiplicativity of the multilinear mapfalong each coordinate.f.map_update_addis the additivity of the multilinear mapfalong each coordinate.f.map_smul_univexpresses the multiplicativity offover all coordinates at the same time, writingf (fun i => c i • m i)as(∏ i, c i) • f m.f.map_add_univexpresses the additivity offover all coordinates at the same time, writingf (m + m')as the sum over all subsetssofιoff (s.piecewise m m').f.map_sumexpressesf (Σ_{j₁} g₁ j₁, ..., Σ_{jₙ} gₙ jₙ)as the sum off (g₁ (r 1), ..., gₙ (r n))whererranges over all possible functions.
See Mathlib.LinearAlgebra.Multilinear.Curry for the currying of multilinear maps.
Implementation notes #
Expressing that a map is linear along the i-th coordinate when all other coordinates are fixed
can be done in two (equivalent) different ways:
- fixing a vector
m : ∀ (j : ι - i), M₁ j.val, and then choosing separately thei-th coordinate - fixing a vector
m : ∀j, M₁ j, and then modifying itsi-th coordinate
The second way is more artificial as the value of m at i is not relevant, but it has the
advantage of avoiding subtype inclusion issues. This is the definition we use, based on
Function.update that allows to change the value of m at i.
Note that the use of Function.update requires a DecidableEq ι term to appear somewhere in the
statement of MultilinearMap.map_update_add' and MultilinearMap.map_update_smul'.
Three possible choices are:
- Requiring
DecidableEq ιas an argument toMultilinearMap(as we did originally). - Using
Classical.decEq ιin the statement ofmap_add'andmap_smul'. - Quantifying over all possible
DecidableEq ιinstances in the statement ofmap_add'andmap_smul'.
Option 1 works fine, but puts unnecessary constraints on the user
(the zero map certainly does not need decidability).
Option 2 looks great at first, but in the common case when ι = Fin n
it introduces non-defeq decidability instance diamonds
within the context of proving map_update_add' and map_update_smul',
of the form Fin.decidableEq n = Classical.decEq (Fin n).
Option 3 of course does something similar, but of the form Fin.decidableEq n = _inst,
which is much easier to clean up since _inst is a free variable
and so the equality can just be substituted.
Multilinear maps over the ring R, from ∀ i, M₁ i to M₂ where M₁ i and M₂ are modules
over R.
- toFun : ((i : ι) → M₁ i) → M₂
The underlying multivariate function of a multilinear map.
- map_update_add' [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i) : self.toFun (Function.update m i (x + y)) = self.toFun (Function.update m i x) + self.toFun (Function.update m i y)
A multilinear map is additive in every argument.
- map_update_smul' [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i) : self.toFun (Function.update m i (c • x)) = c • self.toFun (Function.update m i x)
A multilinear map is compatible with scalar multiplication in every argument.
Instances For
Equations
- MultilinearMap.instFunLikeForall = { coe := fun (f : MultilinearMap R M₁ M₂) => f.toFun, coe_injective' := ⋯ }
Alias of MultilinearMap.map_update_add.
Alias of MultilinearMap.map_update_add.
Earlier, this name was used by what is now called MultilinearMap.map_update_smul_left.
Alias of MultilinearMap.map_update_smul.
Earlier, this name was used by what is now called MultilinearMap.map_update_smul_left.
Alias of MultilinearMap.map_update_smul.
Earlier, this name was used by what is now called MultilinearMap.map_update_smul_left.
Equations
- MultilinearMap.instAdd = { add := fun (f f' : MultilinearMap R M₁ M₂) => { toFun := fun (x : (i : ι) → M₁ i) => f x + f' x, map_update_add' := ⋯, map_update_smul' := ⋯ } }
Equations
- MultilinearMap.instZero = { zero := { toFun := fun (x : (i : ι) → M₁ i) => 0, map_update_add' := ⋯, map_update_smul' := ⋯ } }
Equations
- MultilinearMap.instInhabited = { default := 0 }
Equations
- MultilinearMap.instSMul = { smul := fun (c : S) (f : MultilinearMap R M₁ M₂) => { toFun := fun (m : (i : ι) → M₁ i) => c • f m, map_update_add' := ⋯, map_update_smul' := ⋯ } }
Equations
Coercion of a multilinear map to a function as an additive monoid homomorphism.
Equations
- MultilinearMap.coeAddMonoidHom = { toFun := DFunLike.coe, map_zero' := ⋯, map_add' := ⋯ }
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If f is a multilinear map, then f.toLinearMap m i is the linear map obtained by fixing all
coordinates but i equal to those of m, and varying the i-th coordinate.
Equations
- f.toLinearMap m i = { toFun := fun (x : M₁ i) => f (Function.update m i x), map_add' := ⋯, map_smul' := ⋯ }
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The cartesian product of two multilinear maps, as a multilinear map.
Equations
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Combine a family of multilinear maps with the same domain and codomains M' i into a
multilinear map taking values in the space of functions ∀ i, M' i.
Equations
- MultilinearMap.pi f = { toFun := fun (m : (i : ι) → M₁ i) (i : ι') => (f i) m, map_update_add' := ⋯, map_update_smul' := ⋯ }
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Equivalence between linear maps M₂ →ₗ[R] M₃ and one-multilinear maps.
Equations
- One or more equations did not get rendered due to their size.
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The constant map is multilinear when ι is empty.
Equations
- MultilinearMap.constOfIsEmpty R M₁ m = { toFun := Function.const ((i : ι) → M₁ i) m, map_update_add' := ⋯, map_update_smul' := ⋯ }
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Given a multilinear map f on n variables (parameterized by Fin n) and a subset s of k
of these variables, one gets a new multilinear map on Fin k by varying these variables, and fixing
the other ones equal to a given value z. It is denoted by f.restr s hk z, where hk is a
proof that the cardinality of s is k. The implicit identification between Fin k and s that
we use is the canonical (increasing) bijection.
Equations
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In the specific case of multilinear maps on spaces indexed by Fin (n+1), where one can build
an element of ∀ (i : Fin (n+1)), M i using cons, one can express directly the additivity of a
multilinear map along the first variable.
In the specific case of multilinear maps on spaces indexed by Fin (n+1), where one can build
an element of ∀ (i : Fin (n+1)), M i using cons, one can express directly the multiplicativity
of a multilinear map along the first variable.
In the specific case of multilinear maps on spaces indexed by Fin (n+1), where one can build
an element of ∀ (i : Fin (n+1)), M i using snoc, one can express directly the additivity of a
multilinear map along the first variable.
In the specific case of multilinear maps on spaces indexed by Fin (n+1), where one can build
an element of ∀ (i : Fin (n+1)), M i using cons, one can express directly the multiplicativity
of a multilinear map along the first variable.
If g is a multilinear map and f is a collection of linear maps,
then g (f₁ m₁, ..., fₙ mₙ) is again a multilinear map, that we call
g.compLinearMap f.
Equations
- g.compLinearMap f = { toFun := fun (m : (i : ι) → M₁ i) => g fun (i : ι) => (f i) (m i), map_update_add' := ⋯, map_update_smul' := ⋯ }
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Composing a multilinear map twice with a linear map in each argument is the same as composing with their composition.
Composing the zero multilinear map with a linear map in each argument.
Composing a multilinear map with the identity linear map in each argument.
Composing with a family of surjective linear maps is injective.
Composing a multilinear map with a linear equiv on each argument gives the zero map if and only if the multilinear map is the zero map.
If one adds to a vector m' another vector m, but only for coordinates in a finset t, then
the image under a multilinear map f is the sum of f (s.piecewise m m') along all subsets s of
t. This is mainly an auxiliary statement to prove the result when t = univ, given in
map_add_univ, although it can be useful in its own right as it does not require the index set ι
to be finite.
Additivity of a multilinear map along all coordinates at the same time,
writing f (m + m') as the sum of f (s.piecewise m m') over all sets s.
If f is multilinear, then f (Σ_{j₁ ∈ A₁} g₁ j₁, ..., Σ_{jₙ ∈ Aₙ} gₙ jₙ) is the sum of
f (g₁ (r 1), ..., gₙ (r n)) where r ranges over all functions with r 1 ∈ A₁, ...,
r n ∈ Aₙ. This follows from multilinearity by expanding successively with respect to each
coordinate. Here, we give an auxiliary statement tailored for an inductive proof. Use instead
map_sum_finset.
If f is multilinear, then f (Σ_{j₁ ∈ A₁} g₁ j₁, ..., Σ_{jₙ ∈ Aₙ} gₙ jₙ) is the sum of
f (g₁ (r 1), ..., gₙ (r n)) where r ranges over all functions with r 1 ∈ A₁, ...,
r n ∈ Aₙ. This follows from multilinearity by expanding successively with respect to each
coordinate.
If f is multilinear, then f (Σ_{j₁} g₁ j₁, ..., Σ_{jₙ} gₙ jₙ) is the sum of
f (g₁ (r 1), ..., gₙ (r n)) where r ranges over all functions r. This follows from
multilinearity by expanding successively with respect to each coordinate.
Restrict the codomain of a multilinear map to a submodule.
This is the multilinear version of LinearMap.codRestrict.
Equations
- f.codRestrict p h = { toFun := fun (v : (i : ι) → M₁ i) => ⟨f v, ⋯⟩, map_update_add' := ⋯, map_update_smul' := ⋯ }
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Reinterpret an A-multilinear map as an R-multilinear map, if A is an algebra over R
and their actions on all involved modules agree with the action of R on A.
Equations
- MultilinearMap.restrictScalars R f = { toFun := ⇑f, map_update_add' := ⋯, map_update_smul' := ⋯ }
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Transfer the arguments to a map along an equivalence between argument indices.
The naming is derived from Finsupp.domCongr, noting that here the permutation applies to the
domain of the domain.
Equations
- MultilinearMap.domDomCongr σ m = { toFun := fun (v : ι₂ → M₂) => m fun (i : ι₁) => v (σ i), map_update_add' := ⋯, map_update_smul' := ⋯ }
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MultilinearMap.domDomCongr as an equivalence.
This is declared separately because it does not work with dot notation.
Equations
- MultilinearMap.domDomCongrEquiv σ = { toFun := MultilinearMap.domDomCongr σ, invFun := MultilinearMap.domDomCongr σ.symm, left_inv := ⋯, right_inv := ⋯, map_add' := ⋯ }
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The results of applying domDomCongr to two maps are equal if
and only if those maps are.
If {a // P a} is a subtype of ι and if we fix an element z of (i : {a // ¬ P a}) → M₁ i,
then a multilinear map on M₁ defines a multilinear map on the restriction of M₁ to
{a // P a}, by fixing the arguments out of {a // P a} equal to the values of z.
Given a multilinear map f on (i : ι) → M i, a (decidable) predicate P on ι and
an element z of (i : {a // ¬ P a}) → M₁ i, construct a multilinear map on
(i : {a // P a}) → M₁ i) whose value at x is f evaluated at the vector with ith coordinate
x i if P i and z i otherwise.
The naming is similar to MultilinearMap.domDomCongr: here we are applying the restriction to the
domain of the domain.
For a linear map version, see MultilinearMap.domDomRestrictₗ.
Equations
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The "derivative" of a multilinear map, as a linear map from (i : ι) → M₁ i to M₂.
For continuous multilinear maps, this will indeed be the derivative.
Equations
- f.linearDeriv x = ∑ i : ι, f.toLinearMap x i ∘ₗ LinearMap.proj i
Instances For
Composing a multilinear map with a linear map gives again a multilinear map.
Equations
- g.compMultilinearMap f = { toFun := ⇑g ∘ ⇑f, map_update_add' := ⋯, map_update_smul' := ⋯ }
Instances For
The multilinear version of LinearMap.subtype_comp_codRestrict
The multilinear version of LinearMap.comp_codRestrict
The space of multilinear maps over an algebra over R is a module over R, for the pointwise
addition and scalar multiplication.
LinearMap.compMultilinearMap as an S-linear map.
Equations
- LinearMap.compMultilinearMapₗ S g = { toFun := g.compMultilinearMap, map_add' := ⋯, map_smul' := ⋯ }
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Linear equivalence between linear maps M₂ →ₗ[R] M₃
and one-multilinear maps MultilinearMap R (fun _ : ι ↦ M₂) M₃.
Equations
- One or more equations did not get rendered due to their size.
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The dependent version of MultilinearMap.domDomCongrLinearEquiv.
Equations
- One or more equations did not get rendered due to their size.
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The space of constant maps is equivalent to the space of maps that are multilinear with respect to an empty family.
Equations
- One or more equations did not get rendered due to their size.
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MultilinearMap.domDomCongr as a LinearEquiv.
Equations
- One or more equations did not get rendered due to their size.
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Given a predicate P, one may associate to a multilinear map f a multilinear map
from the elements satisfying P to the multilinear maps on elements not satisfying P.
In other words, splitting the variables into two subsets one gets a multilinear map into
multilinear maps.
This is a linear map version of the function MultilinearMap.domDomRestrict.
Equations
- f.domDomRestrictₗ P = { toFun := fun (z : (i : { a : ι // ¬P a }) → M₁ ↑i) => f.domDomRestrict P z, map_update_add' := ⋯, map_update_smul' := ⋯ }
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One of the components of the iterated derivative of a multilinear map. Given a bijection e
between a type α (typically Fin k) and a subset s of ι, this component is a multilinear map
of k vectors v₁, ..., vₖ, mapping them to f (x₁, (v_{e.symm 2})₂, x₃, ...), where at
indices i in s one uses the i-th coordinate of the vector v_{e.symm i} and otherwise one
uses the i-th coordinate of a reference vector x.
This is multilinear in the components of x outside of s, and in the v_j.
Equations
- One or more equations did not get rendered due to their size.
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The k-th iterated derivative of a multilinear map f at the point x. It is a multilinear
map of k vectors v₁, ..., vₖ (with the same type as x), mapping them
to ∑ f (x₁, (v_{i₁})₂, x₃, ...), where at each index j one uses either xⱼ or one
of the (vᵢ)ⱼ, and each vᵢ has to be used exactly once.
The sum is parameterized by the embeddings of Fin k in the index type ι (or, equivalently,
by the subsets s of ι of cardinality k and then the bijections between Fin k and s).
For the continuous version, see ContinuousMultilinearMap.iteratedFDeriv.
Equations
- f.iteratedFDeriv k x = ∑ e : Fin k ↪ ι, (f.iteratedFDerivComponent e.toEquivRange) fun (i : { a : ι // a ∉ Set.range ⇑e }) => x ↑i
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If f is a collection of linear maps, then the construction MultilinearMap.compLinearMap
sending a multilinear map g to g (f₁ ⬝ , ..., fₙ ⬝ ) is linear in g.
Equations
- MultilinearMap.compLinearMapₗ f = { toFun := fun (g : MultilinearMap R M₁' M₂) => g.compLinearMap f, map_add' := ⋯, map_smul' := ⋯ }
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If f is a collection of linear maps, then the construction MultilinearMap.compLinearMap
sending a multilinear map g to g (f₁ ⬝ , ..., fₙ ⬝ ) is linear in g and multilinear in
f₁, ..., fₙ.
Equations
- MultilinearMap.compLinearMapMultilinear = { toFun := MultilinearMap.compLinearMapₗ, map_update_add' := ⋯, map_update_smul' := ⋯ }
Instances For
Let M₁ᵢ and M₁ᵢ' be two families of R-modules and M₂ an R-module.
Let us denote Π i, M₁ᵢ and Π i, M₁ᵢ' by M and M' respectively.
If g is a multilinear map M' → M₂, then g can be reinterpreted as a multilinear
map from Π i, M₁ᵢ ⟶ M₁ᵢ' to M ⟶ M₂ via (fᵢ) ↦ v ↦ g(fᵢ vᵢ).
Equations
- One or more equations did not get rendered due to their size.
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If one multiplies by c i the coordinates in a finset s, then the image under a multilinear
map is multiplied by ∏ i ∈ s, c i. This is mainly an auxiliary statement to prove the result when
s = univ, given in map_smul_univ, although it can be useful in its own right as it does not
require the index set ι to be finite.
Multiplicativity of a multilinear map along all coordinates at the same time,
writing f (fun i => c i • m i) as (∏ i, c i) • f m.
Given an R-algebra A, mkPiAlgebra is the multilinear map on A^ι associating
to m the product of all the m i.
See also MultilinearMap.mkPiAlgebraFin for a version that works with a non-commutative
algebra A but requires ι = Fin n.
Equations
- MultilinearMap.mkPiAlgebra R ι A = { toFun := fun (m : ι → A) => ∏ i : ι, m i, map_update_add' := ⋯, map_update_smul' := ⋯ }
Instances For
Given an R-algebra A, mkPiAlgebraFin is the multilinear map on A^n associating
to m the product of all the m i.
See also MultilinearMap.mkPiAlgebra for a version that assumes [CommSemiring A] but works
for A^ι with any finite type ι.
Equations
- MultilinearMap.mkPiAlgebraFin R n A = { toFun := fun (m : Fin n → A) => (List.ofFn m).prod, map_update_add' := ⋯, map_update_smul' := ⋯ }
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Given an R-multilinear map f taking values in R, f.smulRight z is the map
sending m to f m • z.
Equations
- f.smulRight z = (LinearMap.id.smulRight z).compMultilinearMap f
Instances For
The canonical multilinear map on R^ι when ι is finite, associating to m the product of
all the m i (multiplied by a fixed reference element z in the target module). See also
mkPiAlgebra for a more general version.
Equations
- MultilinearMap.mkPiRing R ι z = (MultilinearMap.mkPiAlgebra R ι R).smulRight z
Instances For
Equations
- MultilinearMap.instNeg = { neg := fun (f : MultilinearMap R M₁ M₂) => { toFun := fun (m : (i : ι) → M₁ i) => -f m, map_update_add' := ⋯, map_update_smul' := ⋯ } }
Equations
- MultilinearMap.instSub = { sub := fun (f g : MultilinearMap R M₁ M₂) => { toFun := fun (m : (i : ι) → M₁ i) => f m - g m, map_update_add' := ⋯, map_update_smul' := ⋯ } }
Equations
Alias of MultilinearMap.map_update_neg.
Alias of MultilinearMap.map_update_sub.
This calculates the differences between the values of a multilinear map at
two arguments that differ on a finset s of ι. It requires a
linear order on ι in order to express the result.
This expresses the difference between the values of a multilinear map
at two points "close to x" in terms of the "derivative" of the multilinear map at x
and of "second-order" terms.
When ι is finite, multilinear maps on R^ι with values in M₂ are in bijection with M₂,
as such a multilinear map is completely determined by its value on the constant vector made of ones.
We register this bijection as a linear equivalence in MultilinearMap.piRingEquiv.
Equations
- One or more equations did not get rendered due to their size.
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The pushforward of an indexed collection of submodule p i ⊆ M₁ i by f : M₁ → M₂.
Note that this is not a submodule - it is not closed under addition.
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The map is always nonempty. This lemma is needed to apply SubMulAction.zero_mem.
The range of a multilinear map, closed under scalar multiplication.