A multilinear map is a function from several vector spaces to another vector space that is linear in each argument separately. All spaces are taken over the same field, such as the real or complex numbers. Multilinearity extends the notion of a linear map to multiple inputs, but does not generally mean linearity on the product space as a whole. Such maps connect linear algebra with the theory of tensors and tensor products. (math.stanford.edu)
Definition and basic properties
Let (V_1,\ldots,V_n,W) be vector spaces over a field (K), with (n\geq1). A map [ f:V_1\times\cdots\times V_n\longrightarrow W ] is multilinear, or (n)-linear, if, for every position (i), every (u,v\in V_i), and every (a,b\in K), [ f(v_1,\ldots,au+bv,\ldots,v_n)
a f(v_1,\ldots,u,\ldots,v_n) +b f(v_1,\ldots,v,\ldots,v_n). ] The other arguments remain fixed in this identity. Thus each argument respects linear combinations independently. For (n=1), the definition reduces to ordinary linearity; for (n=2), it defines a bilinear map. A scalar-valued multilinear map, with (W=K), is called a multilinear form. (math.stanford.edu)
Repeated application of the definition gives [ f(a_1v_1,\ldots,a_nv_n) =(a_1\cdots a_n)f(v_1,\ldots,v_n). ] Also, the output is zero whenever any argument is zero. Fixing one argument produces a multilinear map in the remaining arguments. Pointwise addition and scalar multiplication make the collection of multilinear maps with specified domain and codomain into a vector space. (math.stanford.edu)
Separate linearity versus joint linearity
The Cartesian product (V_1\times\cdots\times V_n) itself has componentwise vector-space operations. Multilinearity is nevertheless different from linearity with respect to these operations. For a bilinear map (B), [ B(u+u',v+v')
B(u,v)+B(u,v')+B(u',v)+B(u',v'). ] The two mixed terms generally prevent the joint additivity required of a linear map on the product space. Likewise, [ B(cu,cv)=c^2B(u,v), ] rather than (cB(u,v)). (stanford.edu)
The function (B(x,y)=xy) on (K\times K) illustrates the distinction: fixing either input leaves a linear function of the other, although multiplying both inputs changes the output quadratically. More generally, expanding a multilinear map on sums generates one term for every choice of one summand from each argument. This independent distributive behavior is central to multilinear calculations. (math.stanford.edu)
Examples
A real inner product is a bilinear form. In coordinates, a scalar-valued bilinear map on (K^p\times K^q) can be written [ B(x,y)=x^{\mathsf T}Ay ] for a (p\times q) matrix (A). A complex inner product is instead sesquilinear: it is linear in one argument and conjugate-linear in the other, so it is not generally bilinear over the complex field. (math.mit.edu)
The determinant of an (m\times m) matrix is an (m)-linear form when its columns are treated as separate vector arguments. It is also alternating: it vanishes if two columns coincide. Together with the normalization (\det I=1), these properties characterize the determinant. Multilinearity here concerns individual columns, not the matrix viewed as a single input. (web.mit.edu)
Another example is the evaluation pairing [ V^\times V\longrightarrow K,\qquad (\lambda,v)\longmapsto\lambda(v), ] where (V^) is the dual space of linear functionals on (V). Products of functionals similarly give multilinear forms: [ (v_1,\ldots,v_n)\longmapsto \lambda_1(v_1)\cdots\lambda_n(v_n). ] (stanford.edu)
Coordinates and dimension
Suppose each (V_j) has a finite basis (e^{(j)}1,\ldots,e^{(j)}{d_j}). Writing [ v_j=\sum_{i_j=1}^{d_j}x^{(j)}{i_j}e^{(j)}{i_j} ] and expanding yields [ f(v_1,\ldots,v_n)= \sum_{i_1,\ldots,i_n} x^{(1)}{i_1}\cdots x^{(n)}{i_n} f(e^{(1)}{i_1},\ldots,e^{(n)}{i_n}). ] Consequently, (f) is uniquely determined by its values on tuples of basis vectors. Conversely, arbitrary choices of those values define a multilinear map through this formula. (math.mit.edu)
If (W) has finite dimension (r), the space of such maps has dimension [ r,d_1\cdots d_n. ] Choosing an output basis represents the map by an array with one index for each input and one output index. The array depends on the chosen bases; the underlying multilinear map does not. (math.mit.edu)
Tensor products and tensors
The tensor product converts multilinear problems into linear ones. Its universal property states that every multilinear map (f) factors uniquely through a linear map [ \widetilde f:V_1\otimes\cdots\otimes V_n\longrightarrow W, ] satisfying [ \widetilde f(v_1\otimes\cdots\otimes v_n)=f(v_1,\ldots,v_n). ] This correspondence works even for infinite-dimensional vector spaces. (math.stanford.edu)
When the input spaces are finite-dimensional, the space of multilinear maps is canonically isomorphic to [ V_1^\otimes\cdots\otimes V_n^\otimes W. ] In particular, scalar-valued multilinear forms correspond to covariant tensors. The finite-dimensional hypothesis matters for this identification with a tensor product of duals; it is not needed for the universal factorization above. (math.stanford.edu)
Symmetry and differentiation
For a map on (V^n), symmetry means invariance under permutations of the arguments. An alternating map vanishes whenever two arguments coincide and changes sign when two arguments are exchanged. These structures underlie differential forms, whose values at a point are alternating multilinear forms on the tangent space. (math.stanford.edu)
Multilinear maps also describe higher derivatives. For a sufficiently smooth function between finite-dimensional real vector spaces, its (n)th derivative at a point is a symmetric (n)-linear map. Evaluating this map repeatedly on the same increment gives the degree-(n) term in the multivariable Taylor expansion, with factor (1/n!). (math.stanford.edu)