1.9.5 Multilinear Operation Generalization

Multilinear Operation Generalization extends multilinear algebra by abstracting operations across tensor spaces, enabling broader applications in mathematics and physics.

Multilinear Operation Generalization is the extension of linearity, the property that an operation respects addition and scalar multiplication in a single argument, into multilinearity, the property that an operation respects addition and scalar multiplication independently in each of several arguments simultaneously. This generalization is the algebraic mechanism that underlies the entire tensor framework: tensors are, at their core, nothing more than the coefficients of multilinear operations, and every construction that builds tensors from vectors, the tensor product, contraction, symmetrization, is itself an instance of extending a linear operation into a multilinear one.

An operation f on a single vector argument is linear if it satisfies f(αu + βv) = αf(u) + βf(v) for all vectors u, v and scalars α, β. A multilinear operation on several arguments satisfies this same linearity condition separately in each argument, while holding the others fixed. The generalization from a single linear operation to a multilinear one of several arguments is precisely the generalization that transforms vector algebra into tensor algebra.

From One Argument to Several

Linearity in a Single Slot

A linear functional f: V → F assigns a scalar to each vector while respecting addition and scaling; a linear map f: V → W does the same but returns a vector rather than a scalar. Both cases involve exactly one argument, and linearity is a statement about how the operation behaves as that single argument varies.

f (α u + β v) = α f (u) + β f (v)

Independent Linearity in Multiple Slots

A multilinear operation f(v_1, v_2, …, v_k) accepting k vector arguments is required to be linear in each argument individually, when the remaining arguments are held fixed. This is a stronger and more structured condition than linearity in the arguments taken together, since the operation is not required, and generally does not happen, to be linear if all k arguments are varied jointly.

f (v_{1}, \dots, α v_{i} + β w_{i}, \dots, v_{k}) = α f (v_{1}, \dots, v_{i}, \dots, v_{k}) + β f (v_{1}, \dots, w_{i}, \dots, v_{k})

Tensors as Coefficients of Multilinear Operations

The Correspondence Between Tensors and Multilinear Maps

Once a basis is fixed for the underlying vector space, every multilinear operation on k vector arguments can be represented by an array of coefficients, one for each combination of basis vectors chosen for each argument. This coefficient array is precisely a tensor: its number of indices equals the number of arguments of the multilinear operation, and its transformation law under a change of basis is exactly the tensor transformation law, because it is derived directly from how the multilinear operation must remain consistent under the change of basis.

f (e_{i_{1}}, \dots, e_{i_{k}}) = T_{i_{1} \dots i_{k}}

Rank as Argument Count

Because each argument of the multilinear operation contributes one index to the coefficient tensor, the rank of the tensor is exactly the number of arguments the multilinear operation accepts. A bilinear form, accepting two vector arguments, corresponds to a rank-2 tensor; a trilinear form, accepting three, corresponds to a rank-3 tensor. This correspondence is what allows tensor rank to be understood operationally, as a count of independent directions the operation can vary along, rather than merely as a count of indices in an array.

Extending Familiar Bilinear Operations

The Dot Product as a Bilinear Operation

The ordinary dot product u · v is linear in u for fixed v, and linear in v for fixed u, making it a bilinear, and therefore multilinear, operation with two arguments. Its coefficient tensor, in an orthonormal basis, is the identity matrix, a type (0, 2) tensor, confirming that even the most elementary bilinear operation already fits within the tensor framework as a rank-2 tensor.

Determinants as Multilinear Operations

The determinant of a square matrix, viewed as a function of its rows or columns, is linear in each row separately while the others are held fixed, and is additionally alternating, meaning it changes sign whenever two rows are swapped. This makes the determinant a fully antisymmetric multilinear operation of n vector arguments in an n-dimensional space, and its coefficient tensor is the Levi-Civita tensor, a completely antisymmetric rank-n object.

det (v_{1}, \dots, v_{n}) = ε_{i_{1} \dots i_{n}} v_{1}^{i_{1}} \dots v_{n}^{i_{n}}

Constructing Multilinear Operations From Simpler Ones

The Tensor Product as Combination

Given a multilinear operation f on j arguments and a multilinear operation g on k arguments, their tensor product f ⊗ g is a multilinear operation on j + k arguments, defined by applying f to the first j arguments and g to the remaining k, then multiplying the two results. This shows how complex multilinear operations, of arbitrarily many arguments, can always be built by combining simpler multilinear operations of fewer arguments, mirroring exactly how higher-rank tensors are built from lower-rank ones by the tensor product.

Contraction as Argument Reduction

Given a multilinear operation with at least one vector argument and one covector argument, pairing that vector argument with a fixed covector, or summing the covector argument against a chosen basis in a specific paired way, reduces the number of independent arguments by one on each side, producing an operation of two fewer total arguments. This is the operational meaning behind tensor contraction: it is not an arbitrary algebraic manipulation but a genuine reduction in the arity of the underlying multilinear operation.

Significance Within the Generalization Hierarchy

Unifying the Scalar, Vector, and Matrix Cases

Under the multilinear-operation view, a scalar is a multilinear operation of zero arguments, a fixed number requiring no input. A vector, or more precisely a covector, is a multilinear operation of one argument. A matrix, as a bilinear form or as a linear map re-expressed with an extra argument, is a multilinear operation of two arguments. This view supplies a single, coherent reason why the scalar-to-vector-to-matrix-to-tensor hierarchy holds together: each step simply increases the argument count of the underlying multilinear operation by one.

The Operational Meaning of Tensor Algebra

Framing tensors as coefficients of multilinear operations gives every tensor-algebraic construction, addition, tensor product, contraction, symmetrization, an operational meaning in terms of how the underlying multilinear operations combine, reduce, or rearrange their arguments. This is why multilinear operation generalization is regarded as foundational: it is the abstract source from which the entire concrete apparatus of indices, transformation laws, and tensor components is derived.