8.6.4 Tensor Free Index Output Structure

Tensor Free Index Output Structure defines how indices are presented in tensor algebra, clarifying their roles and relationships in mathematical expressions.

Tensor Free Index Output Structure is the arrangement, in rank, shape, and variance, of the resulting tensor object produced by an expression once every dummy index has been summed away, determined entirely by the surviving free indices, their count, their variance as upper or lower, and the order in which they are written.

Determining Rank from Surviving Indices

Counting the Output Slots

The rank of the output object equals the number of distinct free indices remaining after all summations implied by repeated indices have been carried out. An expression reducing to a single free index produces an object with one slot, analogous to a vector or covector, while an expression with two surviving free indices produces an object with two slots, analogous to a matrix.

Y_{i k} = A_{i j} B^{j k}

Here the dummy index (j) vanishes under summation, and the output structure is fixed by the two surviving free indices (i) and (k), giving a rank-two output.

Variance of the Output Slots

Each surviving free index retains its own variance, upper or lower, directly determining whether the corresponding output slot is contravariant or covariant. The output structure therefore records not merely how many free indices survive, but which tier, upper or lower, each one occupies.

Z_{i}^{k} = A_{i} B^{k j} C_{j}

Here the dummy index (j) is summed away, and the output structure is a mixed rank-two object with one lower slot, labeled (i), and one upper slot, labeled (k).

Ordering Within the Output Structure

Positional Convention

Since the relative ordering of free indices generally matters, particularly when the output tensor lacks symmetry, the output structure also records the left-to-right sequence in which the surviving free indices appear, and this ordering must be preserved consistently whenever the resulting object is referenced elsewhere.

Effect of Reordering

Exchanging the written order of two free indices that are not symmetric to one another produces, in general, a different tensor, so the output structure is only fully specified once both the identity of each surviving free index and its position relative to the others has been fixed.

M_{i k} \neq M_{k i}

Output Structure Under Composite Operations

Sequential Contractions

When several contractions are performed in sequence within a longer expression, the output structure is determined only by whichever free indices remain after all dummy pairs, at every stage of the computation, have been eliminated, regardless of how many intermediate tensors were involved.

R_{i} = A_{i j} B^{j k} C_{k}

Both (j) and (k) are dummy indices consumed by successive contractions, leaving (i) as the sole free index, so the output structure of this composite expression is rank one, lower variance.

Practical Summary

Reading off the output structure of any tensor expression is therefore a mechanical procedure: identify every repeated index and remove it as a dummy, then examine what remains, since the surviving free indices, their variance, and their order completely and exclusively determine the rank and type of the resulting tensor.