2.18.3 Tensor Complex Multilinear Operation

Tensor Complex Multilinear Operation is a mapping defined on ordered tuples of vectors drawn from a complex vector space, and possibly its complex dual, that is linear with respect to the field of complex numbers in each argument separately while all other arguments are held fixed. It is the algebraic operation that generates tensors over a complex vector space, playing for complex tensor algebra the same structural role that ordinary multilinear maps play for real tensor algebra, with the added requirement that homogeneity and additivity in every slot must hold for complex scalar multiples, not merely real ones. This distinction matters because a map that is additive and linear over the reals in each argument need not be linear over the complex numbers in that same argument; complex multilinearity is a strictly stronger condition that anchors every subsequent construction, such as complex tensor products, complex symmetric and alternating forms, and complex-valued invariants, to the field C itself rather than to its underlying real structure.

Formal Definition

The Multilinear Map Perspective

Let V_1, ..., V_k be finite-dimensional complex vector spaces. A complex multilinear operation is a map

such that, for every index i and every fixed choice of the remaining arguments, the induced map on the i-th slot alone satisfies

for all complex scalars α, β ∈ C. The output may itself be scalar-valued, giving a complex multilinear form, or vector-valued, giving a multilinear map into another complex vector space; both cases obey the same slot-wise linearity requirement.

Complex Linearity Versus Real Linearity in Each Argument

Any complex vector space can be regarded as a real vector space of twice the dimension by forgetting multiplication by i. A map that is additive and homogeneous under real scalars in each argument, when the spaces are viewed this way, is called real multilinear, and this is a weaker condition than complex multilinearity. A complex multilinear operation must in particular satisfy T(..., i v_i, ...) = i T(..., v_i, ...) in every argument, a requirement that real-multilinear-only maps, such as those built from real and imaginary parts separately, generally fail to satisfy.

Relation to Complex Tensor Spaces

Multilinear Operations as the Source of Tensor Products

Complex multilinear operations on V_1 × ... × V_k correspond, via the universal property of the tensor product, to linear maps on the complex tensor product space V_1 ⊗ ... ⊗ V_k. Concretely, every complex multilinear operation T factors uniquely through the canonical multilinear map into the tensor product, so that T = τ ∘ ⊗, where τ is a linear functional (or linear map) on the tensor product and ⊗ denotes the canonical embedding sending (v_1, ..., v_k) to v_1 ⊗ ... ⊗ v_k. This universal factorization is the mechanism by which complex multilinear operations are converted into elements of, or operators on, complex tensor spaces.

Contraction With Dual Slots

When some arguments are drawn from the dual space V* rather than from V itself, a complex multilinear operation of mixed type produces exactly the component data of a complex tensor of type (p, q), with p slots accepting dual vectors and q slots accepting vectors, evaluated by the natural pairing between V and V* extended complex-bilinearly.

Complex Multilinear Versus Sesquilinear Operations

Distinguishing the Two Kinds of Linearity

A separate but related operation type arises when one or more arguments are required to be conjugate-linear rather than linear, meaning T(..., α v_i, ...) = \bar{α} T(..., v_i, ...) for a conjugated argument. An operation linear in some slots and conjugate-linear in others is called sesquilinear rather than multilinear; the Hermitian inner product on a complex vector space is the standard example, being linear in one argument and conjugate-linear in the other. A tensor complex multilinear operation, by contrast, is required to be linear, never conjugate-linear, in every argument, so sesquilinear forms are explicitly excluded from this notion and instead belong to a distinct algebraic category built from V and its conjugate space \bar{V}.

Why the Distinction Is Preserved in Tensor Algebra

Keeping multilinear and sesquilinear operations separate is necessary because the tensor product functor ⊗ is only compatible with linear, not conjugate-linear, universal factorization. Sesquilinear operations are instead represented using the conjugate vector space \bar{V}, so that a sesquilinear form becomes an honest complex multilinear operation on V × \bar{V}.

Examples

Complex Bilinear Form

The simplest nontrivial case is k = 2: a complex bilinear form B : V × V → C satisfying linearity in both arguments simultaneously, such as B(v, w) = \sum_i v_i w_i in coordinates, with no conjugation applied to either factor.

The Determinant as a Complex Multilinear Operation

For an n-dimensional complex vector space V, the determinant, viewed as a function of n vector arguments expressed in a fixed basis, is a complex multilinear operation that is additionally alternating, changing sign under transposition of any two arguments. This alternating multilinear structure is what makes the determinant the canonical generator of the top exterior power of V.

Diagrammatic Summary

The diagram shows k complex vector arguments feeding into the multilinear operation T, each slot varying linearly with respect to complex scalars independently of the others, and producing a single complex output, the pattern that underlies every complex tensor built by repeated application of such operations.