2.4.1 Tensor Vector Element Identity

Tensor Vector Element Identity explains how tensors interact with vectors, revealing key algebraic relationships in tensor mathematics.

Tensor Vector Element Identity is the criterion by which two elements of a tensor space are recognized as one and the same object, rather than as merely producing similar-looking outputs or descriptions. It fixes what it means for the equation $S = T$ to hold between two tensors, and it is what allows a single tensor to be described in multiple equivalent ways, through different multilinear formulas, different component arrays, or different factorizations, without ambiguity about whether those descriptions denote the same element.

The Identity Criterion

Equality as Multilinear Maps

Let $V$ be a finite-dimensional vector space over a field $F$ , and let $S, T \in T_{s}^{r} (V)$ . Since a tensor is intrinsically a multilinear map, two tensors are identical precisely when they agree on every possible input:

S = T \Leftrightarrow S (ω^{1}, \dots, v_{s}) = T (ω^{1}, \dots, v_{s})

for every choice of covector arguments $ω^{k} \in V^{*}$ and vector arguments $v_{k} \in V$ .

Reduction to Componentwise Equality

Because a tensor is completely determined by its action on basis vectors and dual basis covectors, this universal condition reduces, once a basis of $V$ is fixed, to a finite check: equality holds if and only if every component agrees,

S = T \Leftrightarrow S_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}} = T_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}}

for every admissible index combination, replacing an infinite family of conditions with a finite array comparison.

Basis Independence of Identity

Why the Criterion Does Not Depend on the Basis

Multilinear equality, stated as agreement on every input, refers to no basis at all, so it is manifestly basis-independent. The componentwise criterion inherits this independence indirectly: if components agree in one basis, the linear transformation law relating components in any second basis applies identically to both $S$ and $T$ , forcing their components to agree in that second basis as well.

Practical Significance

This independence is what permits identity to be verified in whichever basis is most convenient. A computation may switch between coordinate systems freely while checking whether two tensors coincide, without risk of a false positive or false negative arising from the choice of basis.

Identity and the Zero Tensor

Uniqueness of the Zero Element

The zero tensor, the multilinear map sending every input tuple to $0$ , is the unique element satisfying $T + 0 = T$ for all $T$ . If a second element $0^{'}$ also satisfied this identity, then setting $T = 0$ shows $0 + 0^{'} = 0$ , while setting $T = 0^{'}$ shows $0^{'} + 0 = 0^{'}$ , and combining these with commutativity gives $0 = 0^{'}$ .

Testing Identity via Subtraction

Two tensors $S$ and $T$ are identical exactly when their difference is the zero tensor:

S = T \Leftrightarrow S - T = 0

reducing the identity question to a single computation rather than a pairwise comparison across every component.

Identity Under Different Representations

Non-Uniqueness of Factorization for Simple Tensors

A simple tensor $T = v \otimes w$ can be written using many different factor pairs; replacing $v$ with $α v$ and $w$ with $(1 / α) w$ for nonzero $α$ leaves the tensor unchanged as an element, since the multilinear map computed from either factorization agrees on every input. Element identity depends only on the resulting multilinear map, never on which factorization produced it.

Uniqueness of the Component Representation

By contrast, once a basis is fixed, the array of components representing a given tensor is unique: a tensor cannot be assigned two different component arrays relative to the same basis, since the components are defined directly as the values of the tensor on specific basis vector and dual basis covector combinations, and those values are fixed by the tensor itself.

Identity and Linear Combinations

Equality of Linear Combinations

Since every element of $T_{s}^{r} (V)$ has a unique expansion as a linear combination of the basis tensors formed from tensor products of basis vectors and dual basis covectors, two linear combinations denote the same element if and only if their coefficients agree termwise. This uniqueness is what allows identity of tensors to be checked by direct comparison of coefficients whenever both tensors are expressed relative to the same tensor product basis.

Distinguishing Identity from Numerical Coincidence

Element identity requires agreement on the full multilinear map, not merely agreement on a single output value or a single component. Two distinct tensors may share some components or agree on some inputs while still differing overall, so identity must always be verified against the complete set of components or, equivalently, against every possible input to the multilinear map.