13.10.3 Tensor Matrix Multiplication Output Index Pair

Tensor Matrix Multiplication Output Index Pair is the pair of free indices, one contravariant and one covariant, that survive the contraction underlying matrix multiplication and together constitute the index structure of the resulting rank-two tensor. It identifies specifically which slot of each input factor contributes to the output, distinguishing these surviving indices from the shared index that is summed away and disappears entirely from the final expression.

Conceptual Basis

Composition of Free Indices From Each Factor

Matrix multiplication contracts exactly one index from each of the two factor tensors while leaving one additional index on each factor untouched. The output index pair is formed by taking the untouched contravariant index from the first factor together with the untouched covariant index from the second factor.

Origin in Row and Column Structure

In the traditional description of matrix multiplication, the output index pair corresponds to the row index of the first matrix and the column index of the second matrix, reflecting that the product's entries are organized by these two surviving labels once the shared intermediate index has been summed out.

Determining the Type of the Result

Because the output index pair consists of exactly one contravariant and one covariant index, the resulting tensor is always of type $$(1,1)$$, regardless of the specific dimensions involved, confirming that matrix multiplication preserves the mixed rank-two character of its inputs.

Formal Description

Identifying the Output Index Pair

For mixed tensors $$A_j^i$$ and $$B_k^j$$, the shared index $$j$$ is contracted, leaving $$i$$ from the first factor and $$k$$ from the second factor as the output index pair, giving the resulting tensor:

with $$i$$ as the contravariant member of the output index pair and $$k$$ as the covariant member.

Dimension of Each Output Index

The contravariant member of the output index pair ranges over the dimension associated with the first factor's contravariant index, while the covariant member ranges over the dimension associated with the second factor's covariant index, and these two dimensions need not match each other even though the contracted shared index dimension must match across both factors.

Transformation Behavior of the Output Pair

Under a change of basis, the contravariant member of the output index pair transforms via the Jacobian matrix associated with the first factor's original space, and the covariant member transforms via the inverse Jacobian associated with the second factor's original space, since the contracted index no longer contributes any transformation dependence.

Properties

Independence of the Two Members

The two members of the output index pair originate from entirely separate tensors and separate vector spaces in general, meaning they can be reindexed, renamed, or transformed independently of one another, subject only to each retaining consistency with its own originating space.

Role in Determining Compatibility for Further Contraction

Any subsequent contraction applied to the resulting tensor must draw its indices from the output index pair, since the shared index used in the original multiplication is no longer available, and the specific variances of the two remaining indices determine which further pairings are valid.

Preservation Under Chained Multiplication

When the resulting tensor participates in a further matrix multiplication with a third tensor, one member of its output index pair becomes the new shared index for that subsequent contraction, while the other member persists as part of the output index pair of the extended product.

Practical Considerations

Reading Off Dimensions From the Output Pair

The dimensions of the output index pair directly determine the size of the resulting matrix, with the contravariant member's dimension giving the number of rows and the covariant member's dimension giving the number of columns of the product, in the conventional matrix representation.

Relevance to Non-Square Products

When the two factors involved are not square, the output index pair may have members of differing dimension, producing a rectangular resulting tensor whose shape is fully determined by the dimensions inherited from the respective free indices of the two factors.

Consistency Checks in Longer Expressions

In an expression involving several chained tensor multiplications, tracking the output index pair at each stage provides a straightforward way to confirm that later contractions in the chain are dimensionally valid, since each subsequent contraction must match against the appropriate member of the preceding output index pair.