9.8.5 Tensor Basis Tensor Component Support

Tensor Basis Tensor Component Support explains how tensors are built and expressed using bases, enabling precise algebraic representation and manipulation.

Tensor Basis Tensor Component Support is the identification of exactly which basis tensors participate, with a nonzero coefficient, in the expansion of a given tensor, distinguishing the basis elements that genuinely contribute to reconstructing that tensor from the basis elements whose coefficient vanishes and which may therefore be omitted from the expansion without altering the result; it names the subset of a tensor basis actually needed to represent one particular tensor, as opposed to the full basis needed to represent every tensor of that type in general.

Defining Component Support Precisely

The Set of Index Combinations With Nonzero Coefficient

Given a tensor expanded in a basis as a sum over basis tensors weighted by components, the component support of that tensor is the set of index combinations for which the corresponding component is nonzero; every basis tensor whose index combination lies outside this set contributes nothing to the expansion and can be dropped from it.

support (T) = {(i, j) : T^{i j} \neq 0}

An Equivalent, Reduced Expansion

Once the component support is identified, the tensor's expansion can be rewritten as a sum restricted to only those index combinations in the support, producing an equivalent expression that omits every basis tensor known in advance to make no contribution.

T = \sum_{(i, j) \in support (T)} T^{i j} e_{i} \otimes e_{j}

Why Component Support Depends on the Basis

A Tensor's Support Is Not an Intrinsic Property Alone

Because component assignment itself depends on the chosen basis, the same tensor can have a large component support in one basis and a much smaller one in another; a tensor with many nonzero entries relative to one basis may become sparse, or even have a support consisting of a single index combination, once expressed in a more suitable basis.

Diagonalization as an Extreme Case of Reduced Support

Choosing a basis in which a tensor's component support is as small as possible — for instance, a basis in which a symmetric tensor's off-diagonal entries all vanish — is precisely the goal underlying diagonalization procedures, which seek out a basis minimizing the component support of a specific tensor of interest.

Practical Uses of Identifying Component Support

Reducing Computational Work

Once the component support of a tensor is known, calculations involving that tensor — contractions, comparisons, or further tensor products — can be restricted to the index combinations within the support, skipping terms known in advance to contribute nothing and reducing the amount of computation required.

Recognizing Special Structural Properties

A tensor whose component support is confined to a particular pattern of indices — for instance, only to combinations where all indices coincide — reveals a structural property of that tensor, such as being purely diagonal in the given basis, that might not be evident from the abstract tensor alone without examining its support.

Diagram of Component Support

Consequences of Component Support

It Enables Efficient Representation of Tensors

Recognizing that a tensor's component support may be much smaller than the full set of possible index combinations allows the tensor to be recorded and manipulated using only its nonzero entries, avoiding the storage and processing of entries already known to vanish.

It Highlights the Basis-Dependence of Apparent Sparsity

Because component support changes with the chosen basis, a tensor appearing "sparse," with a small support, in one basis carries no implication that it remains sparse in another; any claim of sparsity must be understood as relative to the specific basis in which the component support was determined.