13.10 Tensor Matrix Multiplication Contraction Case

Tensor Matrix Multiplication Contraction Case is the specific instance of tensor contraction in which two rank-two mixed tensors are combined by summing over a single shared index, one contravariant on the first factor and one covariant on the second, reproducing exactly the standard operation of matrix multiplication as it is known in linear algebra. It identifies matrix multiplication not as a separate operation but as a particular realization of the general contraction mechanism applied to tensors of the simplest nontrivial rank capable of representing linear maps.

Conceptual Basis

Matrices as Rank-Two Mixed Tensors

A matrix representing a linear map between vector spaces corresponds to a rank-two mixed tensor, with one contravariant index associated with the output space and one covariant index associated with the input space. The matrix multiplication contraction case arises when two such tensors are joined by contracting the output index of one against the input index of the other.

Composition of Linear Maps

Multiplying two matrices corresponds to composing the linear maps they represent, and the contraction underlying this multiplication is precisely what encodes this composition in index notation, with the contracted index representing the intermediate space through which the composed map passes.

Position Within the Broader Family of Contractions

Among the various named cases of tensor contraction, the matrix multiplication case is characterized by involving exactly two tensors, each of rank two, joined through exactly one shared index, distinguishing it from trace contraction, which acts within a single tensor, and from more general inner products, which may involve tensors of arbitrary matching rank.

Formal Description

Index Notation

For mixed tensors $$A_j^i$$ and $$B_k^j$$, the matrix multiplication contraction case is written:

with the index $$j$$ summed over its full range and absent from the resulting tensor $$P_k^i$$.

Dimensional Compatibility

The contracted index must range over the same dimension on both factors, corresponding to the familiar requirement that the number of columns of the first matrix equal the number of rows of the second, since the summation defining the product would otherwise be undefined.

Explicit Componentwise Form

Written without the summation convention, each entry of the resulting tensor is:

where $$m$$ is the shared dimension of the contracted index.

Properties

Non-Commutativity

Because the contraction pairs a specific covariant index on the first factor with a specific contravariant index on the second, reversing the order of the two tensors generally links different index slots and produces a different resulting tensor, reflecting the well-known non-commutativity of matrix multiplication.

Associativity Across Chained Products

When three or more mixed tensors are multiplied in sequence, each adjacent contraction can be carried out independently, and the overall result does not depend on the order in which the individual contractions are performed, mirroring the associativity of matrix multiplication.

Rank Preservation

Unlike trace contraction, which reduces rank by consuming both indices of a single tensor, the matrix multiplication contraction case preserves the total rank of the resulting object at two, since each factor contributes exactly one free index to the final tensor.

Relationship to Other Contraction Cases

Reduction to the Trace Case

Applying the matrix multiplication contraction case to a tensor multiplied by itself, followed by a further trace contraction over the remaining mixed pair, produces a scalar equal to the sum of squared eigenvalues or related spectral invariants, linking this case directly to the trace contraction case.

Generalization to Higher-Rank Products

The same contraction pattern extends to tensors of higher rank by contracting one designated index pair while leaving the remaining indices on each factor free, situating the matrix multiplication contraction case as the simplest instance of this more general composition pattern.

Connection to the Inner Product Case

When the free indices of the resulting product tensor are themselves contracted against a metric or against another vector, the matrix multiplication contraction case becomes a component of a larger inner product computation, illustrating how these named contraction cases combine within more elaborate tensor expressions.