12.9 Tensor Evaluation Operation

Tensor Evaluation Operation is the operation by which a tensor, understood as a multilinear map, is applied to an appropriate collection of vectors and covectors to produce a single scalar output, realizing the tensor's role as a function rather than merely as an array of components.

Formal Definition

Tensor as a Multilinear Map

A tensor $$A$$ of type $$(p,q)$$ can be regarded as a multilinear map taking $$p$$ covectors and $$q$$ vectors as input and producing a scalar as output:

where each $${\omega }^k$$ is a covector from the dual space and each $$v_k$$ is a vector from the underlying vector space, and $$c$$ is the resulting scalar.

Componentwise Formula for Evaluation

Given a basis, the evaluation operation can be computed explicitly by contracting the tensor's components with the components of the input vectors and covectors:

with implied summation over each repeated index appearing once as an upper index and once as a lower index.

Multilinearity of Evaluation

Linearity in Each Argument Separately

Evaluation is linear in every one of its arguments individually. Fixing all but one input and varying that single input linearly, whether it is a vector or a covector, produces an output that scales and adds linearly with respect to that input:

Distinction from Ordinary Function Evaluation

Because of this multilinearity, tensor evaluation behaves quite differently from evaluating an arbitrary nonlinear function, since the output responds proportionally and additively to changes in each argument, holding the others fixed.

Basis Independence of Evaluation

The Scalar Output Does Not Depend on the Basis Used

Although the componentwise formula for evaluation appears to depend on a chosen basis, the resulting scalar $$c$$ is the same regardless of which basis is used to carry out the computation, since the transformation laws of the tensor's components and the vectors and covectors involved cancel out exactly in the contraction.

Consistency Check Across Bases

If the same vectors and covectors, along with the same tensor, are re-expressed in a different basis and the evaluation is recomputed there, the resulting scalar remains identical, confirming that evaluation captures an intrinsic property of the tensor and its arguments rather than an artifact of coordinates.

Role Within Tensor Algebra

Foundation for Interpreting Tensors as Functions

Evaluation is what allows a tensor to be understood not merely as a static array of numbers but as an active object capable of processing input vectors and covectors to yield meaningful scalar quantities, such as an inner product, a determinant contribution, or a physical measurement.

Relation to Contraction

The evaluation operation is closely related to tensor contraction, since evaluating a tensor on specific vectors and covectors can be viewed as contracting the tensor's indices against the components of those inputs.

Illustration