4.9.3 Tensor Scalar Valued Evaluation

Tensor Scalar Valued Evaluation is the computational procedure by which the scalar output of a multilinear map is actually produced from a chosen basis, the tensor's components in that basis, and the coordinate representations of the supplied arguments, reducing the abstract act of "applying $T$ to a tuple" to a concrete finite sum that can be carried out arithmetically. It is the practical, algorithmic counterpart to the abstract definition of a scalar-valued multilinear map, and it is what turns tensor theory into something directly computable.

Formal Definition

The Evaluation Formula

Given a type $(r, s)$ tensor $T$ with components $T^{i_1 \cdots i_r}{\phantom{i_1 \cdots i_r} j_1 \cdots j_s}$ relative to a basis ${e_1, \ldots, e_n}$ of $V$ and dual basis ${e^1, \ldots, e^n}$ of $V^{}$, and given arguments $\phi^{(1)}, \ldots, \phi^{(r)} \in V^{}$ and $v{(1)}, \ldots, v_{(s)} \in V$ expanded in coordinates as $\phi^{(a)} = \phi^{(a)}{i_a} e^{i_a}$ and $v{(b)} = v_{(b)}^{j_b} e_{j_b}$, the scalar valued evaluation is

a sum, in principle running over all $n^{r+s}$ combinations of the indices $i_1, \ldots, i_r, j_1, \ldots, j_s$, each ranging from $1$ to $n$, of a product of one component of $T$ and one coordinate from each argument.

Derivation From Slotwise Linearity

This formula is not an additional assumption but a direct consequence of applying slotwise additivity and homogeneity once per slot: expanding the first argument in coordinates and using linearity in that slot alone reduces the evaluation to a sum over the first index, and repeating this process for every remaining slot produces the full multi-index sum above.

Practical Aspects of the Evaluation

Reduction to Finitely Many Multiplications and Additions

Regardless of how the vector spaces involved are described abstractly, once a basis is fixed, evaluating $T$ on any tuple requires only a finite, predictable number of scalar multiplications and additions, namely one multiplication per term and $n^{r+s} - 1$ additions to sum the terms, making the evaluation procedure tractable even for tensors of substantial arity.

Sparsity and Efficient Evaluation

If many of the tensor's components are zero, as is common for tensors with special symmetry (such as antisymmetric or diagonal metrics), the sum in the evaluation formula can be restricted to only the nonzero component terms, considerably reducing the number of arithmetic operations required in practice compared to the naive full sum over all $n^{r+s}$ index combinations.

Order Independence of Summation

Because ordinary field addition and multiplication are commutative and associative, the order in which the terms of the evaluation sum are computed and combined does not affect the final scalar result, allowing the evaluation to be carried out in any convenient order, including reordering to exploit sparsity or to group terms by shared indices.

Relation to Other Tensor Operations

Evaluation as Repeated Slot Fixing

The evaluation procedure can be understood as applying slot fixing sequentially to every slot of the tensor, one basis-expanded argument at a time, until no free slots remain and only a bare scalar is left; each partial sum accumulated during the process corresponds to an intermediate tensor of successively lower arity.

Evaluation Underlies Contraction

Contraction of a tensor against itself over a pair of matched indices is computed by an evaluation-like summation restricted to the diagonal terms where the contracted upper and lower indices coincide, showing that the general scalar valued evaluation formula and the specific operation of contraction share the same underlying summation mechanism, differing only in which indices are summed and under what constraint.

Summary of Key Points

• Scalar valued evaluation is the concrete summation procedure that computes a tensor's numeric output from its components and the coordinates of its arguments.
• The evaluation formula follows directly from applying slotwise additivity and homogeneity once per argument slot.
• The procedure requires only a finite number of multiplications and additions, and can be simplified considerably when many components are zero.
• Because field arithmetic is commutative and associative, the summation can be performed in any convenient order.
• The evaluation procedure underlies both the sequential act of slot fixing and the summation mechanism used to compute contraction.