4.6.5 Tensor Multilinearity Verification

Tensor Multilinearity Verification is the procedure by which a candidate map on a product of vector spaces is confirmed, or refuted, as genuinely multilinear by systematically testing the slotwise additivity and slotwise homogeneity conditions in every argument slot. It is the practical, checkable counterpart to the abstract definition of a tensor as a multilinear map, providing a finite and mechanical set of tests that either certify a map as a legitimate tensor or exhibit a concrete counterexample showing it is not.

The Verification Procedure

Slot-by-Slot Testing

Given a candidate map $T : V_1 \times \cdots \times V_k \to F$, verification proceeds by isolating each slot index $i$ in turn, fixing an arbitrary but specific fixed argument context for the remaining $k-1$ slots, and testing two conditions on the resulting single-variable function:

If both hold for arbitrary $u, w$ in that slot's space and arbitrary scalar $c$, and this is confirmed for every slot in the argument structure, the map is certified multilinear. By the principle of independent slot variation, these $k$ checks may be performed in any order and do not interact.

Sufficiency of Basis-Level Testing

Because linearity in a finite-dimensional space is fully determined by its action on a basis, verification can be reduced to checking additivity and homogeneity only on basis vectors of each slot, rather than on arbitrary elements: if the two conditions hold when $u$ and $w$ range over a spanning set, they extend automatically to all of $V_i$ by the linear extension property, making the verification task finite even though the vector spaces themselves may be infinite in cardinality.

Common Failure Modes Detected by Verification

Detecting Non-Additivity

A map that fails additivity in some slot typically arises from an operation that does not distribute over vector sums, such as taking a norm, a maximum, or another nonlinear function of one argument. Verification catches this by exhibiting a specific pair $u, w$ in that slot for which the two sides of the additivity equation differ.

Detecting Non-Homogeneity

A map that fails homogeneity in some slot typically arises from an operation with the wrong scaling degree, such as squaring an argument before use, or applying an absolute value. Verification catches this by exhibiting a specific scalar $c$ and vector $u$ for which scaling the input does not scale the output by the same factor.

Detecting Cross-Slot Contamination

A subtler failure occurs when a proposed formula for $T$ secretly couples two slots together in a nonlinear way, such as multiplying two argument vectors' entries in an unintended manner. Verification exposes this by fixing all slots except the two suspected of contamination and checking that varying each independently still produces linear behavior, revealing the coupling as a violation in one of the two isolated checks.

Practical Considerations

Verification on Components

For a map already given in explicit index or component form, verification reduces to confirming that the formula for $T$ is, term by term, a homogeneous degree-one polynomial expression in each set of index variables corresponding to a single slot, with no cross terms mixing two different slots' coordinates raised to any power other than the first, and no terms independent of a slot's coordinates that should instead scale with them.

Verification as a Prerequisite for Tensor Operations

Establishing multilinearity through verification is a prerequisite before applying any tensor operation, such as computing components, performing contraction, or checking symmetry, since all of those operations implicitly assume the slotwise linearity property holds; applying them to a non-multilinear map produces meaningless or inconsistent results.

Summary of Key Points

• Multilinearity verification systematically tests slotwise additivity and homogeneity in each argument slot of a candidate map.
• Testing can be restricted to basis vectors in each slot, since linear behavior on a basis extends automatically to the whole space.
• Common failures include nonlinear operations within a single slot and unintended coupling between two different slots.
• Verification on explicit component formulas reduces to confirming each slot's coordinates appear homogeneously to the first power, with no improper cross terms.
• Successful verification is a necessary prerequisite before applying any tensor operation that relies on the slotwise linearity property.