Alternating multilinear map

In multilinear algebra, an alternating multilinear map (also known as an alternating multilinear form) is a multilinear map f: V^k \to W between vector spaces over a field, where V and W are the domain and codomain spaces, respectively, that vanishes identically whenever any two input vectors are equal, i.e., f(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = 0 whenever v_i = v_j for all i \neq j and all choices of vectors.^[1] This condition implies that f is antisymmetric under the exchange of any two arguments: f(\dots, v_i, \dots, v_j, \dots) = -f(\dots, v_j, \dots, v_i, \dots).^[2] Equivalently, for any permutation \sigma of the indices, f(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = (\operatorname{sgn} \sigma) f(v_1, \dots, v_k), where \operatorname{sgn} \sigma is the sign of the permutation.^[3] Alternating multilinear maps form a vector subspace of the space of all multilinear maps, denoted A^k(V; W), and play a central role in the construction of exterior algebras and differential forms.^[1] For a finite-dimensional vector space V of dimension n over a field of characteristic not 2, the dimension of A^k(V; \mathbb{R}) (or more generally over the base field) is \binom{n}{k}, corresponding to the basis elements of the k-th exterior power \bigwedge^k V.^[2] There is a canonical isomorphism between A^k(V; W) and the space of linear maps \operatorname{Hom}(\bigwedge^k V, W), establishing the universal property that any alternating multilinear map factors uniquely through the exterior product.^[1] A quintessential example is the determinant function \det: V^n \to K, where K is the base field, which is alternating multilinear and normalized such that \det(I) = 1 for the identity matrix; this property uniquely determines the determinant up to scalar multiple.^[3] In three dimensions, the cross product v \times w can be viewed through the alternating bilinear map associated with the volume form, satisfying u \cdot (v \times w) = \det(u, v, w).^[1] These maps generalize to higher-degree forms in differential geometry, where they underpin concepts like oriented volumes, Stokes' theorem, and integration over manifolds.^[2]

Basic Concepts

Multilinear Maps

A multilinear map, also known as a k-linear map, is a function f: V_1 \times \cdots \times V_k \to F between the Cartesian product of vector spaces V_1, \dots, V_k over a field F and the field F itself (or more generally to another vector space W) that is linear in each argument separately when the remaining arguments are held fixed.^[1] This means that for each i = 1, \dots, k, the map v_i \mapsto f(v_1, \dots, v_{i-1}, v_i, v_{i+1}, \dots, v_k) is a linear transformation from V_i to F, for any fixed choice of vectors in the other spaces.^[4] Explicitly, multilinearity implies additivity and homogeneity in each slot:

f(v_1, \dots, v_i + w_i, \dots, v_k) = f(v_1, \dots, v_i, \dots, v_k) + f(v_1, \dots, w_i, \dots, v_k)

and

f(v_1, \dots, \lambda v_i, \dots, v_k) = \lambda f(v_1, \dots, v_i, \dots, v_k)

for all vectors v_j \in V_j, w_i \in V_i, and scalars \lambda \in F.^[4] These properties extend naturally to the case where the codomain is any vector space W, allowing multilinear maps to serve as building blocks for more complex algebraic structures. The set of all k-linear maps from V_1 \times \cdots \times V_k to F itself forms a vector space under pointwise addition and scalar multiplication.^[1] The tensor product provides a universal construction for multilinear maps. Specifically, there is a natural isomorphism between the space of multilinear maps \mathrm{Mult}(V_1 \times \cdots \times V_k, F) and the dual space of the tensor product (V_1 \otimes \cdots \otimes V_k)^*. This follows from the universal property of the tensor product: given any multilinear map g: V_1 \times \cdots \times V_k \to W, there exists a unique linear map \tilde{g}: V_1 \otimes \cdots \otimes V_k \to W such that \tilde{g}(v_1 \otimes \cdots \otimes v_k) = g(v_1, \dots, v_k) for all v_i \in V_i, with the bilinear (or multilinear) map \phi: V_1 \times \cdots \times V_k \to V_1 \otimes \cdots \otimes V_k given by \phi(v_1, \dots, v_k) = v_1 \otimes \cdots \otimes v_k.^[1] This property characterizes the tensor product up to unique isomorphism and underscores its role in linearizing multilinear constructions.^[4] Multilinear maps emerged in the late 19th century as part of the foundational work in tensor calculus developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita, whose 1900 paper systematized the absolute differential calculus and introduced tensorial concepts essential for modern differential geometry.^[5] Alternating multilinear maps represent a special case where additional symmetry conditions are imposed.^[1]

Alternating Maps

An alternating multilinear map is a special type of multilinear map that vanishes identically whenever any two input vectors are equal. Specifically, given a vector space V over a field F, a k-linear map f: V^k \to F (or more generally to W) is alternating if f(v_1, \dots, v_k) = 0 whenever v_i = v_j for some i \neq j.^[1] Over fields of characteristic not equal to 2, this condition implies that f is antisymmetric under the exchange of any two distinct arguments: 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 \neq j and all v_1, \dots, v_k \in V.^[6]^[7] The antisymmetry condition extends to arbitrary permutations because the symmetric group S_k is generated by adjacent transpositions. Thus, f(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = \operatorname{sgn}(\sigma) f(v_1, \dots, v_k) for any permutation \sigma \in S_k, where \operatorname{sgn}(\sigma) is the sign of \sigma. Such maps are called skew-symmetric.^[7]^[6] Equivalently, for a general multilinear map g: V^k \to F, the associated alternating map \operatorname{Alt}(g) is given by the alternatization formula:

\operatorname{Alt}(g)(v_1, \dots, v_k) = \frac{1}{k!} \sum_{\sigma \in S_k} \operatorname{sgn}(\sigma) \, g(v_{\sigma(1)}, \dots, v_{\sigma(k)}).

This construction ensures that \operatorname{Alt}(g) is skew-symmetric and vanishes on repeated arguments. Over fields of characteristic zero, multilinear maps that vanish whenever any two arguments are repeated coincide with skew-symmetric multilinear maps.^[1]^[8] Such maps are commonly defined on the Cartesian product V^k, where V is finite-dimensional, with codomain the base field F (or generally W).^[7] In practice, alternating maps are often normalized so that their value on an ordered basis of V aligns with determinant-like behavior, for instance, evaluating to 1 on the standard basis vectors.^[6] This normalization facilitates their role in algebraic constructions while preserving the antisymmetric structure.^[7]

Properties

Fundamental Properties

A fundamental property of alternating multilinear maps is their vanishing on linearly dependent sets of vectors. Specifically, for an alternating k-linear map f: V^k \to F over a field F, where V is a vector space, f(v_1, \dots, v_k) = 0 whenever the vectors v_1, \dots, v_k \in V are linearly dependent.^[7] By definition, f vanishes whenever any two arguments coincide. In fields of characteristic not equal to 2, f is antisymmetric: 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 i < j.^[1] For linear dependence, express one vector as a linear combination of others; multilinearity then reduces the value to a sum of terms where repeated vectors appear, each vanishing by the above, yielding zero overall.^[3] The space of alternating k-linear maps \mathrm{Alt}^k(V; F) from a finite-dimensional vector space V of dimension n to F has dimension \binom{n}{k}.^[9] This dimension arises because \mathrm{Alt}^k(V; F) is isomorphic to the k-th exterior power of the dual space \wedge^k V^*, whose basis consists of the wedge products of ordered subsets of the dual basis elements.^[1] Consequently, \dim \mathrm{Alt}^k(V; F) = 0 if k > n, reflecting that no non-zero alternating map exists on more than n vectors in an n-dimensional space.^[7] Given a basis \{e_1, \dots, e_n\} of V, there exists essentially a unique alternating k-linear map up to scalar multiple that takes the value 1 on the ordered tuple (e_1, \dots, e_k), corresponding to the basis element e_1 \wedge \cdots \wedge e_k.^[1] This uniqueness stems from the universal mapping property of the exterior algebra, which identifies alternating maps with linear functionals on \wedge^k V.^[9] In the special case k = n, this reduces to the determinant function normalized on the basis, unique up to scaling.^[3] Alternating multilinear maps provide a measure of signed k-dimensional volumes in V. For vectors v_1, \dots, v_k \in V, the value f(v_1, \dots, v_k) represents the oriented volume of the parallelepiped they span, with the sign determined by the orientation relative to a fixed basis.^[1] In Euclidean spaces like \mathbb{R}^n, this aligns with the absolute value giving the unsigned volume, while the alternation encodes the handedness.^[3] For k = n, it coincides with the determinant, scaling the standard volume form.^[7]

Alternatization

The alternatization operator, often denoted Alt, is a linear map that projects the space of multilinear maps into the subspace of alternating multilinear maps. For a k-linear map f: V^k \to F over a field F, where V is a vector space, the alternatization is defined by averaging f over all permutations of its arguments, weighted by the sign of each permutation.^[10]^[11] The explicit formula is given by

\begin{aligned} \text{Alt}(f)(v_1, \dots, v_k) &= \frac{1}{k!} \sum_{\sigma \in S_k} \operatorname{sgn}(\sigma) \, f(v_{\sigma(1)}, \dots, v_{\sigma(k)}), \end{aligned}

where S_k is the symmetric group on k elements and \operatorname{sgn}(\sigma) \in \{ \pm 1 \} is the sign of the permutation \sigma.^[10]^[11] This construction ensures that \text{Alt}(f) vanishes whenever any two arguments v_i = v_j for i \neq j, thereby yielding an alternating map.^[10] The operator is idempotent, meaning \text{[Alt](/page/Alt)}(\text{[Alt](/page/Alt)}([f](/page/For))) = \text{[Alt](/page/Alt)}([f](/page/For)) for any multilinear f, as it acts as a projection onto the subspace of alternating multilinear maps. To see this, apply the definition to \text{[Alt](/page/Alt)}([f](/page/For)):

\text{[Alt](/page/Alt)}(\text{[Alt](/page/Alt)}([f](/page/For)))(v_1, \dots, v_k) = \frac{1}{k!} \sum_{\tau \in S_k} \operatorname{sgn}(\tau) \, \text{[Alt](/page/Alt)}([f](/page/For))(v_{\tau(1)}, \dots, v_{\tau(k)}).

Substituting the formula for \text{[Alt](/page/Alt)}([f](/page/For)) yields a double sum over \sigma, \tau \in S_k:

\frac{1}{(k!)^2} \sum_{\tau, \sigma} \operatorname{sgn}(\tau) \operatorname{sgn}(\sigma) f(v_{\tau(\sigma(1))}, \dots, v_{\tau(\sigma(k))}).

Let \pi = \tau \circ \sigma \in S_k; as \sigma, \tau range over S_k, so does \pi, and \operatorname{sgn}(\tau) \operatorname{sgn}(\sigma) = \operatorname{sgn}(\pi). For each fixed \pi, there are exactly k! pairs (\sigma, \tau) with \tau \circ \sigma = \pi, so the double sum simplifies to

\frac{1}{k!} \sum_{\pi \in S_k} \operatorname{sgn}(\pi) f(v_{\pi(1)}, \dots, v_{\pi(k)}) = \text{[Alt](/page/Alt)}([f](/page/For))(v_1, \dots, v_k).

This confirms idempotence and the projection property.^[10]^[11] The kernel of Alt consists of those multilinear maps f for which \text{Alt}(f) = 0, which occurs precisely when the signed sum over permutations vanishes for all inputs. The image of Alt is exactly the space \text{Alt}^k(V; F) of all k-linear alternating maps from V^k to F.^[10] This operator factors through the k-th exterior power \bigwedge^k V, in the sense that alternating maps correspond to linear functionals on \bigwedge^k V, and Alt induces the universal construction mapping tensor powers to exterior powers via antisymmetrization.^[11] The definition is well-defined over fields F of characteristic not equal to $2, where \operatorname{sgn}(\sigma)

distinguishes even and odd permutations. In [characteristic](/page/Characteristic) &#36;2

, however, -1 = 1, so \operatorname{sgn}(\sigma) = 1 for all \sigma, making alternatization coincide with symmetrization (the average without signs). In such cases, the operator fails to produce genuinely alternating maps unless the original f already satisfies additional antisymmetry conditions.^[7]^[10]

Examples and Applications

Basic Examples

A fundamental example of an alternating bilinear map arises on the vector space \mathbb{R}^2, where the map f: \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R} defined by f((x_1, y_1), (x_2, y_2)) = x_1 y_2 - x_2 y_1 computes the signed area of the parallelogram spanned by the input vectors.^[12] This map is multilinear, as it is linear in each argument separately, and alternating because swapping the inputs negates the value: f((x_2, y_2), (x_1, y_1)) = x_2 y_1 - x_1 y_2 = -f((x_1, y_1), (x_2, y_2)).^[12] For instance, applying f to the standard basis vectors e_1 = (1,0) and e_2 = (0,1) yields f(e_1, e_2) = 1, while f(e_2, e_1) = -1.^[12] In three dimensions, an alternating trilinear map on \mathbb{R}^3 is given by the scalar triple product f(u, v, w) = \det\begin{bmatrix} u & v & w \end{bmatrix}, which measures the signed volume of the parallelepiped spanned by u, v, w.^[1] This map is multilinear and alternating, as permuting the arguments by an odd permutation changes the sign, while an even permutation preserves it.^[1] For example, if u = (1,0,0), v = (0,1,0), and w = (0,0,1), then f(u,v,w) = 1; swapping v and w gives f(u,w,v) = -1.^[1] The map vanishes on collinear vectors, such as when w lies in the span of u and v, reflecting zero volume.^[1] A key property of any alternating multilinear map f: V^k \to F (for k \geq 2) is that it vanishes whenever two arguments are identical: f(v, \dots, v, \dots, w, \dots) = 0.^[3] For the trilinear case on \mathbb{R}^3, this means f(v,v,w) = 0 for any v, w \in \mathbb{R}^3, as the inputs fail to span a full-dimensional volume.^[1] Alternating multilinear maps can be represented as skew-symmetric tensors, where the components A_{i_1 \dots i_k} satisfy A_{\pi(i_1) \dots \pi(i_k)} = \operatorname{sgn}(\pi) A_{i_1 \dots i_k} for permutations \pi.^[7] Such tensors arise via alternatization of a general multilinear map T, given by A_{i_1 \dots i_k} = \frac{1}{k!} \sum_{\sigma \in S_k} \operatorname{sgn}(\sigma) T_{\sigma(i_1) \dots \sigma(i_k)}.^[7] For the bilinear wedge product of covectors a, b \in (\mathbb{R}^2)^*, the components follow this skew-symmetric form: (a \wedge b)(v_1, v_2) = a(v_1)b(v_2) - a(v_2)b(v_1).^[4]

Determinant Function

The determinant function serves as the canonical example of an alternating multilinear map in the context of square matrices. For an n \times n matrix A with real entries, the determinant \det: (\mathbb{R}^n)^n \to \mathbb{R} is defined as an n-linear map that is linear in each column vector separately.^[2] It is alternating, meaning that swapping any two columns negates the value of the determinant, and it satisfies the normalization condition \det(I_n) = 1, where I_n is the n \times n identity matrix.^[2] This structure captures the signed volume of the parallelepiped spanned by the column vectors.^[13] The explicit form of the determinant is given by the Leibniz formula:

\det(A) = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma(i)},

where S_n is the symmetric group of permutations of \{1, \dots, n\}, and \operatorname{sgn}(\sigma) is the sign of the permutation \sigma (+1 for even permutations and -1 for odd).^[2] This formula arises as the alternatization of the permanent, which is the analogous multilinear map without the sign factor: the alternatization operator A_n f = \frac{1}{n!} \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \sigma f applied to the permanent yields the determinant, confirming its alternating property.^[2] A key result is the uniqueness of the determinant among such maps. Any alternating n-linear map f: (\mathbb{R}^n)^n \to \mathbb{R} satisfying f(e_1, \dots, e_n) = 1, where \{e_1, \dots, e_n\} is the standard basis, coincides with the determinant when evaluated on column vectors of a matrix.^[13] This uniqueness follows from the one-dimensionality of the space of alternating n-forms on \mathbb{R}^n.^[2] The multilinearity of the determinant is further illustrated by cofactor expansion, a recursive formula that expands along the j-th row (or column):

\det(A) = \sum_{i=1}^n (-1)^{i+j} a_{ij} \det(M_{ij}),

where M_{ij} is the (n-1) \times (n-1) minor obtained by deleting row i and column j from A.^[14] This expansion leverages multilinearity by treating the entries a_{ij} as scalars multiplying the determinants of the minors, allowing computation through linear combinations in each row or column.^[14] The determinant map \det: \mathrm{GL}(n, \mathbb{R}) \to \mathbb{R}^\times is a surjective group homomorphism whose kernel is the special linear group \mathrm{SL}(n, \mathbb{R}), consisting of all invertible n \times n matrices with determinant 1.^[15] Computationally, evaluating the determinant of an n \times n dense matrix over the reals requires O(n^\omega) arithmetic operations in the worst case, where \omega \approx 2.3713 is the exponent of matrix multiplication; practical algorithms achieve this bound using fast matrix multiplication techniques.^[16]