2.18.1 Tensor Complex Scalar Compatibility

Tensor Complex Scalar Compatibility explores how complex scalars interact with tensor structures, ensuring consistent mathematical behavior across algebraic operations.

Tensor Complex Scalar Compatibility is the requirement that every scalar appearing in the construction, transformation, and manipulation of tensors on a complex vector space V belongs to the field C and interacts with vectors, covectors, and tensor products according to the complex field axioms, while additionally specifying, slot by slot, whether a given multilinear or sesquilinear operation is linear or conjugate-linear in that scalar. Unlike the real case, where "compatible with scalar multiplication" has only one meaning, the complex case forks into two distinct compatibility regimes — ordinary complex bilinearity and Hermitian sesquilinearity — and a large part of what this compatibility concept governs is keeping those two regimes from being conflated.

Two Regimes of Complex Compatibility

Ordinary Complex Bilinear Compatibility

For the plain tensor product V ⊗_C W and multilinear operations built from it, scalar compatibility works exactly as in the real case but with a ∈ C: a scalar factors freely out of any slot without conjugation.

T (\dots a v \dots) = a \cdot T (\dots v \dots)

with no conjugate bar appearing anywhere. Every ordinary type (p, q) complex tensor, built from copies of V and V*, obeys this unconjugated form of compatibility in all p + q slots.

Sesquilinear Compatibility for Hermitian Structures

Hermitian forms and inner products on a complex vector space instead require sesquilinear compatibility: linear in one argument and conjugate-linear in the other. For a Hermitian form h, the standard convention is:

h (a v, w) = a \cdot h (v, w), h (v, a w) = \bar{a} \cdot h (v, w)

Scalar compatibility for this type of pairing means that any scalar pulled from the second slot must be conjugated before it can be combined with a scalar pulled from the first slot, so h(av, bw) = a ̄b · h(v, w), not ab · h(v, w).

Consequences for Component Arithmetic

Complex-Valued Components

The components of an ordinary complex type (p, q) tensor, once a complex coordinate system is fixed, are elements of C. Scaling the tensor by a complex number a scales every component uniformly by a, without conjugation, exactly mirroring the real case but with complex arithmetic:

(a, T)^{i} = a T^{i}

Conjugate-Compatibility in Hermitian Component Arrays

For a Hermitian form represented by a matrix H of complex components, scalar compatibility together with the Hermitian symmetry condition H_{ij} = ̄{H_{ji}} forces every diagonal entry H_{ii} to be real, since a complex number equal to its own conjugate must have zero imaginary part. This is the component-level signature of sesquilinear scalar compatibility and has no analogue in the ordinary bilinear regime.

Compatibility Under Contraction

Contraction of an ordinary complex tensor behaves exactly as in the real case: a complex scalar factored out before contraction equals the same scalar factored out after, by associativity and commutativity of C. When a contraction instead pairs a vector index against a conjugated dual index — as happens when computing a Hermitian norm h(v, v) = ̄{v^i} H_{ij} v^j — scalar compatibility must be checked slot by slot, tracking which factors are conjugated and which are not.

Why the Fork Matters

Preventing Ill-Defined Expressions

Failing to track which regime applies leads to expressions that are not well defined: applying ordinary bilinear compatibility to a Hermitian pairing, for instance, would predict h(av, aw) = a^2 h(v, w) for every complex a, whereas the correct sesquilinear result is h(av, aw) = |a|^2 h(v, w), a real nonnegative multiple rather than an arbitrary complex one. This distinction is exactly what allows Hermitian forms to produce a well-defined, real-valued, nonnegative squared length.

Real Numbers Recovered From Complex Compatibility

Sesquilinear scalar compatibility is precisely the mechanism that lets a complex vector space carry a positive-definite real-valued norm despite C itself having no order: h(v, v) is always real by the sesquilinear compatibility rule, and its nonnegativity (when h is a genuine inner product) restores exactly the ordered-comparison capability that C alone lacks.

Diagrammatic Summary

Complex scalar compatibility governs two related but distinct rules for moving a scalar through a multilinear expression, and identifying which rule applies is essential to correctly manipulating complex tensors and Hermitian forms alike.