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Multilinear map

In mathematics, a multilinear map is a function f: V_1 \times V_2 \times \cdots \times V_k \to W between vector spaces over a field that is linear in each argument V_i separately, while holding the arguments in the other spaces fixed.^[1] This means that for any fixed vectors in the other spaces, f behaves as a linear transformation with respect to each individual input vector, satisfying additivity and homogeneity in that variable.^[2] Multilinear maps form the cornerstone of multilinear algebra, which extends classical linear algebra to functions depending on multiple vector spaces.^[1] The space of all k-linear maps from V_1 \times \cdots \times V_k to W itself constitutes a vector space, and multilinear maps are intimately connected to tensor products via the universal property: every multilinear map factors uniquely through the tensor product space V_1 \otimes \cdots \otimes V_k.^[1] Tensors, often defined as multilinear maps from products of a vector space and its dual to the base field, provide a coordinate-free framework for these constructions, with the space of type-(r,s) tensors on an n-dimensional space having dimension n^{r+s}.^[2] Beyond pure algebra, multilinear maps underpin key concepts in geometry and analysis, such as the determinant as an alternating multilinear form on \mathbb{R}^n, which measures volume scaling under linear transformations by |\det A|.^[2] They also arise in differential geometry through pullback operations on tensor fields and in applications like change-of-variables formulas in multiple integrals via the Jacobian.^[2] In modern contexts, multilinear maps extend to computational and cryptographic settings, enabling constructions like tensor networks for data analysis and approximate realizations in lattice-based cryptography.^[3]^[4]

Fundamentals

Definition

A multilinear map is a function f: V_1 \times \cdots \times V_k \to W between vector spaces V_1, \dots, V_k and W over a common field F, which is linear in each argument separately.^[1] Specifically, for each i = 1, \dots, k, scalars \lambda, \mu \in F, and vectors u, v \in V_i with all other arguments fixed, the map satisfies

f(v_1, \dots, v_{i-1}, \lambda u + \mu v, v_{i+1}, \dots, v_k) = \lambda f(v_1, \dots, v_{i-1}, u, v_{i+1}, \dots, v_k) + \mu f(v_1, \dots, v_{i-1}, v, v_{i+1}, \dots, v_k).

^[5] This separate linearity in each variable distinguishes multilinear maps from general functions on the product space, as it requires additivity and homogeneity only when varying one input at a time while holding others constant.^[2] A special case arises when the codomain W is the base field F itself, yielding a multilinear form, which is a scalar-valued multilinear map f: V_1 \times \cdots \times V_k \to F.^[6] These forms generalize linear functionals (the case k=1) to multiple inputs. The vector spaces involved need not be finite-dimensional, though this assumption simplifies many subsequent constructions.^[1]

Historical Context

The origins of multilinear maps trace back to 19th-century advancements in linear algebra, notably Hermann Grassmann's 1844 publication Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics), where he developed the exterior algebra to handle multilinear forms for volumes and oriented subspaces without relying on coordinates.^[7] Grassmann's extension theory provided an early framework for multilinear mappings, emphasizing geometric intuitions over analytic methods.^[8] Concurrently, Arthur Cayley's pioneering work on invariant theory from the 1840s onward explored multilinear algebraic forms in the context of binary quadratic forms and their symmetries under linear transformations, forging overlooked links to group actions that influenced later algebraic developments. In the early 20th century, Élie Cartan advanced the formalization of multilinear maps through his synthesis of Lie groups and differential geometry, particularly in his 1899 thesis and subsequent works on continuous groups, where tensor fields emerged as sections of multilinear bundles on manifolds.^[9] Cartan's exterior differential systems, building on Grassmann's ideas, positioned alternating multilinear maps—known as differential forms—as fundamental tools for analyzing geometric structures. The practical impetus for abstract treatments of multilinear maps came from physics, as Marcel Grossmann acquainted Albert Einstein with tensor calculus in 1912–1913, facilitating the 1915 formulation of general relativity, where tensors served as multilinear maps to encode gravitational effects invariantly across coordinate systems. This application underscored the need for coordinate-free multilinear algebra in curved spaces.^[10] Following World War II, multilinear maps became central to algebraic geometry and representation theory, with post-1950s developments by figures like Alexander Grothendieck integrating them into sheaf theory and tensor categories for studying varieties and symmetries.^[11] These eras marked a shift toward functorial and categorical perspectives, solidifying multilinear maps as a cornerstone of modern pure mathematics.^[12]

Examples

Basic Examples

A fundamental example of a bilinear map is the standard dot product on \mathbb{R}^2, defined by \langle u, v \rangle = u_1 v_1 + u_2 v_2 for u = (u_1, u_2) and v = (v_1, v_2), which maps \mathbb{R}^2 \times \mathbb{R}^2 to \mathbb{R}.^[13] This function is linear in the first argument when the second is fixed: for scalars \alpha, \beta and vectors u, u', \langle \alpha u + \beta u', v \rangle = \alpha \langle u, v \rangle + \beta \langle u', v \rangle, and similarly linear in the second argument when the first is fixed.^[14] An example of a trilinear map is the scalar triple product f(u, v, w) = u \cdot (v \times w) on \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}, where \cdot denotes the dot product and \times the cross product.^[15] This map is linear in each argument separately when the others are held constant; for instance, fixing v and w, f(\alpha u + \beta u', v, w) = \alpha f(u, v, w) + \beta f(u', v, w), with analogous properties for the other variables.^[2] Although this map is alternating, its multilinearity follows directly from the linearity of the dot and cross products. In general, a k-linear map on finite-dimensional vector spaces, such as V_1 \times \cdots \times V_k \to \mathbb{R}, is linear in each of its k arguments when the remaining k-1 are fixed.^[16] For the case k=2, consider a map g: V \times V \to \mathbb{R} on a two-dimensional space V with basis \{e_1, e_2\}; explicitly, g(e_1, e_1) = 1, g(e_1, e_2) = 0, g(e_2, e_1) = 0, g(e_2, e_2) = 1, which extends bilinearly to all pairs and matches the dot product in the standard basis.^[14] To verify multilinearity for any such map, fix all but one argument and confirm linearity in the varying one: additivity and homogeneity in scalars must hold for each position independently.^[16] This stepwise check leverages the definition of multilinearity as iterated bilinearity.^[2]

Geometric and Algebraic Examples

In three-dimensional Euclidean space \mathbb{R}^3, the cross product operation defines a bilinear map from \mathbb{R}^3 \times \mathbb{R}^3 to \mathbb{R}^3.^[17] For vectors u = (u_1, u_2, u_3) and v = (v_1, v_2, v_3), the cross product u \times v yields the vector (u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1), which is linear in each argument separately while preserving the vector space structure.^[17] This bilinearity follows from the distributive properties of the operation over vector addition and scalar multiplication in each component.^[17] The determinant function provides another geometric example of a multilinear map, serving as an alternating trilinear form on the space of row vectors in \mathbb{R}^3.^[18] Specifically, for three vectors v_1, v_2, v_3 \in \mathbb{R}^3, the determinant \det(v_1, v_2, v_3) is linear in each vector argument, measuring the signed volume of the parallelepiped they span.^[19] This multilinearity ensures that scaling any input vector by a scalar multiplies the output by that scalar, independently for each position.^[19] In algebraic contexts, multiplication within the quaternion algebra \mathbb{H} exemplifies a bilinear map.^[20] Quaternions, as elements of \mathbb{R}^4 with basis \{1, i, j, k\}, undergo multiplication defined by rules such as i^2 = j^2 = k^2 = -1 and ij = k, resulting in a bilinear operation over \mathbb{R} that is linear in each quaternion factor.^[21] This structure extends the complex numbers, enabling representations of rotations in three dimensions through non-commutative bilinear products.^[20] Geometrically, multilinear maps capture oriented volumes and areas by generalizing scalar products to higher dimensions.^[22] For instance, the magnitude of the cross product |u \times v| represents the area of the parallelogram spanned by u and v with an orientation determined by the right-hand rule, while the determinant extends this to the signed volume in higher dimensions.^[17] Such maps thus quantify parallelepiped volumes, providing a foundation for measuring oriented subspaces in vector spaces.^[22]

Representations

Coordinate Representation

To represent a multilinear map f: V_1 \times \cdots \times V_k \to W in coordinates, select bases \{e_{j,i}\}_{i=1}^{\dim V_j} for each vector space V_j (j = 1, \dots, k) and \{b_m\}_{m=1}^{\dim W} for the codomain W. The map f is then fully determined by its coefficients A_{i_1 \cdots i_k}^m = f(e_{1,i_1}, \dots, e_{k,i_k}) expressed in the basis of W, or more precisely, the components where f(e_{1,i_1}, \dots, e_{k,i_k}) = \sum_m A_{i_1 \cdots i_k}^m b_m.^[23]^[2] For arbitrary inputs v_j = \sum_i v_{j,i} e_{j,i} in each V_j, the multilinearity of f yields the explicit coordinate formula:

f(v_1, \dots, v_k) = \sum_{i_1, \dots, i_k, m} A_{i_1 \cdots i_k}^m \, v_{1,i_1} \cdots v_{k,i_k} \, b_m.

This summation expands the map as a linear combination over all multi-indices, with the coefficients A_{i_1 \cdots i_k}^m capturing the action on basis elements.^[23]^[2] The space of all k-linear maps from \prod_{j=1}^k V_j to W forms a vector space whose dimension equals the product of the dimensions of the domain factors times the dimension of the codomain; specifically, if each V_j is isomorphic to a space V of dimension n and \dim W = p, then this dimension is n^k p. This follows from the number of independent coefficients needed to specify the map, one for each combination of basis inputs and output basis projections.^[2] As a concrete example, consider a trilinear map f: \mathbb{R}^2 \times \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R} (so W = \mathbb{R} with basis \{1\}) defined by f(u,v,w) = u_1 v_1 w_1 + u_2 v_2 w_2, where subscripts denote standard basis coordinates with e_1 = (1,0) and e_2 = (0,1). The coefficients are A_{111} = f(e_1,e_1,e_1) = 1, A_{222} = f(e_2,e_2,e_2) = 1, and A_{i_1 i_2 i_3} = 0 otherwise. For inputs u = (u_1, u_2), v = (v_1, v_2), w = (w_1, w_2), the formula gives f(u,v,w) = 1 \cdot u_1 v_1 w_1 + 1 \cdot u_2 v_2 w_2, matching the definition and illustrating the summation over non-zero coefficients.^[23]

Relation to Tensor Products

A multilinear map f: V_1 \times \cdots \times V_k \to W between vector spaces over a field corresponds bijectively to a linear map F: V_1 \otimes \cdots \otimes V_k \to W through the tensor product space, satisfying the universal property: for any such multilinear f, there exists a unique linear F such that f(v_1, \ldots, v_k) = F(v_1 \otimes \cdots \otimes v_k) for all v_i \in V_i.^[1]^[24] This bijection ensures that the tensor product V_1 \otimes \cdots \otimes V_k acts as the "universal" object capturing all multilinear behaviors into W, with the canonical multilinear map \phi: V_1 \times \cdots \times V_k \to V_1 \otimes \cdots \otimes V_k given by \phi(v_1, \ldots, v_k) = v_1 \otimes \cdots \otimes v_k.^[1] To establish this correspondence, the tensor product is constructed as a quotient space that enforces multilinearity: start with the free vector space on V_1 \times \cdots \times V_k, then quotient by the subspace generated by relations like (v_1 + \lambda u_1, v_2, \ldots, v_k) - (v_1, v_2, \ldots, v_k) - \lambda (u_1, v_2, \ldots, v_k) and permutations thereof, yielding linearity in each argument separately.^[1] The induced map F inherits linearity from f's multilinearity via the bilinearity of the tensor operation, and uniqueness follows from the spanning property of elementary tensors v_1 \otimes \cdots \otimes v_k in the tensor product.^[24] For multilinear forms (where W is the base field), this relation exhibits contravariance: such a form f: V_1 \times \cdots \times V_k \to K corresponds to an element of the dual tensor product (V_1^* \otimes \cdots \otimes V_k^*), or equivalently to the dual of the tensor product (V_1 \otimes \cdots \otimes V_k)^*, via the isomorphism V_i^* \otimes W \cong \Hom(V_i, W) extended multilinearly.^[1] This algebraic equivalence is confirmed dimensionally: if each V_i has dimension n_i, then \dim(V_1 \otimes \cdots \otimes V_k) = \prod_{i=1}^k n_i, matching the dimension of the space of multilinear maps to a one-dimensional W, which aligns with the coordinate representation where tensor components serve as coefficients of multilinear maps in chosen bases.^[1]^[24]

Applications

Multilinear Functions on Matrices

In the context of linear algebra over a field \mathbb{F}, an n \times n matrix A can be viewed as an ordered tuple of its row vectors ( \mathbf{r}_1, \dots, \mathbf{r}_n ), where each \mathbf{r}_i \in \mathbb{F}^n. Thus, A belongs to the product space (\mathbb{F}^n)^n. A multilinear function on such matrices is a map D: (\mathbb{F}^n)^n \to \mathbb{F} that is linear in each row argument separately.^[1]^[19] The multilinearity condition implies that for row vectors \mathbf{a}_1, \dots, \mathbf{a}_n, \mathbf{b}_1 \in \mathbb{F}^n and scalars \lambda, \mu \in \mathbb{F},

D(\lambda \mathbf{a}_1 + \mu \mathbf{b}_1, \mathbf{a}_2, \dots, \mathbf{a}_n) = \lambda D(\mathbf{a}_1, \mathbf{a}_2, \dots, \mathbf{a}_n) + \mu D(\mathbf{b}_1, \mathbf{a}_2, \dots, \mathbf{a}_n),

with analogous properties holding for linearity in each of the other row positions i = 2, \dots, n. This extends the standard notion of multilinearity to the specific setting where the domain is structured as a product of copies of \mathbb{F}^n, allowing the function to depend on the matrix entries in a way that respects the vector space structure of each row.^[24]^[19] To express such a function explicitly, consider the standard basis for \mathbb{F}^n, consisting of the unit row vectors \hat{\mathbf{e}}_j for j = 1, \dots, n, where \hat{\mathbf{e}}_j has a 1 in the j-th position and 0s elsewhere. Any multilinear function D is uniquely determined by its values on tuples of these basis vectors, i.e., D(\hat{\mathbf{e}}_{j_1}, \dots, \hat{\mathbf{e}}_{j_n}). For a general matrix A = (a_{ik}) with rows \mathbf{r}_i = (a_{i1}, \dots, a_{in}), multilinearity yields the expansion

D(A) = \sum_{j_1=1}^n \cdots \sum_{j_n=1}^n c_{j_1 \dots j_n} a_{1 j_1} a_{2 j_2} \cdots a_{n j_n},

where the coefficients c_{j_1 \dots j_n} = D(\hat{\mathbf{e}}_{j_1}, \dots, \hat{\mathbf{e}}_{j_n}) capture the function's behavior on the basis. This form highlights how D acts as a weighted sum of products of matrix entries, one from each row.^[1]^[24] The space of all such multilinear functions on (\mathbb{F}^n)^n forms a vector space of dimension n^n, as it is spanned by the n^n basis functions corresponding to the monomials a_{1 j_1} \cdots a_{n j_n} for each multi-index (j_1, \dots, j_n). This dimension reflects the freedom in choosing the coefficients without additional constraints from multilinearity alone, emphasizing the structural richness of these maps beyond any specific normalization or symmetry requirements.^[19]^[24]

Role in Determinants and Volumes

One key application of multilinear maps arises in the computation of the determinant of a square matrix, which can be characterized as an alternating multilinear map on the rows or columns of the matrix. Specifically, for an n \times n matrix A with rows \mathbf{r}_1, \dots, \mathbf{r}_n, the determinant \det(A) is given by D(\mathbf{r}_1, \dots, \mathbf{r}_n), where D: (\mathbb{R}^n)^n \to \mathbb{R} is the unique alternating multilinear map satisfying D(\mathbf{e}_1, \dots, \mathbf{e}_n) = 1 for the standard basis vectors \mathbf{e}_i of the identity matrix.^[25] This characterization ensures that the determinant is linear in each row separately while vanishing whenever two rows are identical, capturing the essential geometric and algebraic properties of the matrix.^[26] To illustrate, consider the $2 \times 2 case where A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}. The determinant expands as \det(A) = a_{11}a_{22} - a_{12}a_{21}, which demonstrates multilinearity: fixing the second row, \det(A) is linear in the first row (a_{11}, a_{12}), and similarly for the second row. This expansion arises directly from the alternating multilinear form, where the subtraction accounts for the sign change upon swapping rows.^[27] Geometrically, in \mathbb{R}^n, the absolute value of the determinant |\det(A)| measures the signed n-dimensional volume of the parallelepiped spanned by the row vectors \mathbf{r}_1, \dots, \mathbf{r}_n, with the sign indicating orientation. This volume interpretation follows from the multilinearity, which aligns with how volumes scale under linear combinations in each direction, and the alternation, which enforces zero volume for degenerate (coplanar) spans.^[28] A non-alternating analog is the permanent of A, defined as \operatorname{per}(A) = \sum_{\sigma \in S_n} \prod_{i=1}^n a_{i,\sigma(i)}, which is a multilinear map in the rows without the sign changes, thus providing an unsigned measure of "volume" in combinatorial contexts, such as counting perfect matchings.^[29] This multilinear structure of the determinant extends to integration theory, where it plays a central role in the change-of-variables formula for multiple integrals: for a diffeomorphism \phi: U \to V in \mathbb{R}^n, \int_V f(\mathbf{y}) \, d\mathbf{y} = \int_U f(\phi(\mathbf{x})) |\det(D\phi(\mathbf{x}))| \, d\mathbf{x}, scaling volumes under coordinate transformations.^[30]

Properties

Fundamental Properties

A multilinear map f: V_1 \times \cdots \times V_k \to W, where each V_i and W are vector spaces over a field F, is defined to be linear in each argument separately. This means that for each fixed index i (with $1 \leq i \leq k) and fixed vectors in the other slots, the map v \mapsto f(v_1, \dots, v_{i-1}, v, v_{i+1}, \dots, v_k) is a linear transformation from V_i to W.^[2]^[31] Linearity in each slot implies two core properties: additivity and homogeneity. Additivity states that for any vectors u, v \in V_i and fixed arguments in the other slots,

f(\dots, u + v, \dots) = f(\dots, u, \dots) + f(\dots, v, \dots).

Homogeneity follows similarly: for any scalar \lambda \in F and vector u \in V_i,

f(\dots, \lambda u, \dots) = \lambda f(\dots, u, \dots).

These hold independently for each slot i. A direct consequence of homogeneity is that the map vanishes when any argument is the zero vector: f(\dots, 0, \dots) = 0, since setting \lambda = 0 yields zero regardless of the other inputs. Thus, if any input space V_i = \{0\}, the multilinear map must be the zero map.^[2]^[32]^[31] Multilinearity is preserved under composition with linear maps in individual slots. Specifically, if g: U \to V_i is linear and f is multilinear, then the composed map f \circ (\mathrm{id}_{V_1}, \dots, \mathrm{id}_{V_{i-1}}, g, \mathrm{id}_{V_{i+1}}, \dots, \mathrm{id}_{V_k}) remains multilinear, as linearity in the i-th slot composes with g's linearity to yield overall linearity in each argument. This property underscores the functorial nature of multilinear maps and facilitates their use in constructing tensor products via universal mapping properties.^[32]^[31]

Extensions to Alternating and Symmetric Maps

Symmetric multilinear maps are a special class of multilinear maps that remain invariant under any permutation of their input arguments. Specifically, for a k-linear map f: V^k \to W between vector spaces, symmetry means that f(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = f(v_1, \dots, v_j, \dots, v_i, \dots, v_k) for all i, j and all vectors v_1, \dots, v_k \in V.^[33] These maps form a subspace of the full space of multilinear maps and are closely related to symmetric tensors, which arise as the quotient of the tensor product V^{\otimes k} by the subspace generated by differences like v_i \otimes v_j - v_j \otimes v_i.^[34] The universal property of the symmetric algebra ensures that every symmetric multilinear map factors uniquely through the symmetric power \Sym^k(V).^[34] In contrast, alternating multilinear maps, also known as skew-symmetric maps, change sign under odd permutations of their arguments. For such a map f: V^k \to W, swapping two distinct arguments yields f(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = -f(v_1, \dots, v_j, \dots, v_i, \dots, v_k), and more generally, f(\sigma \cdot (v_1, \dots, v_k)) = \sgn(\sigma) f(v_1, \dots, v_k) for any permutation \sigma.^[33] This antisymmetry implies that alternating maps vanish if any two arguments are identical, and they form the basis for the exterior algebra, where the wedge product \wedge: V^k \to \Lambda^k(V) quotients out the relations enforcing alternation.^[1] The space of alternating k-forms on a vector space V of dimension n has dimension \binom{n}{k}, reflecting the choice of k linearly independent vectors up to ordering.^[35] Similarly, the dimension of the space of symmetric k-multilinear forms is \binom{n + k - 1}{k}, corresponding to the number of multi-indices in a basis expansion.^[35] A canonical example of an alternating multilinear map is the determinant function on n \times n matrices, which can be viewed as \det: V^n \to K for an n-dimensional space V over field K, satisfying alternation and normalizing to 1 on a basis.^[28] This ties into volume computations but highlights the role of alternation in ensuring orientability and uniqueness up to scalar. Beyond linear algebra, alternating multilinear maps underpin differential k-forms on manifolds, which are integrated over oriented submanifolds to define measures invariant under coordinate changes.^[36]

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[PDF] Multilinear Algebra - Alexander Rhys Duncan
Jan 23, 2023 · As in the bilinear case, multilinear maps form a vector space and the symmetric, skew-symmetric, and alternating forms are subspaces of the ...
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https://duncan.math.sc.edu/s23/math742/notes/lin_alg.pdf
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[PDF] Lecture 3.2: Symmetric and skew-symmetric multilinear forms
The determinant is actually a property of a linear map, not a matrix. In this section, we will define and study the determinant in this more abstract context.
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[PDF] Differential Forms - MIT Mathematics
Feb 1, 2019 · One of the goals of this text on differential forms is to legitimize this interpretation of equa- tion (1) in 𝑛 dimensions and in fact, ...