13.10.2 Tensor Matrix Multiplication Summed Column Row Pair

Tensor Matrix Multiplication Summed Column Row Pair combines indices to compute elements by summing products of corresponding row and column entries.

Tensor Matrix Multiplication Summed Column Row Pair is the specific relationship between the column index of the first factor and the row index of the second factor that is identified as the shared index summed over in matrix multiplication, named explicitly in terms of the classical row-and-column description of matrices rather than in abstract contravariant-covariant index language. It designates the same underlying contracted index discussed in general contraction terms, but frames it through the traditional vocabulary of rows and columns familiar from elementary matrix arithmetic.

Conceptual Basis

Rows and Columns as Index Roles

In the classical description of a matrix, entries are organized by a row label and a column label. When two matrices are multiplied, the column label of the first matrix and the row label of the second matrix are identified with one another and summed, and this identified pair is what constitutes the summed column row pair.

Correspondence With Tensor Index Variance

The column index of the first factor corresponds to its covariant index in tensor notation, while the row index of the second factor corresponds to its contravariant index, so the summed column row pair is the same object as the shared index discussed in tensor contraction terms, expressed instead through matrix terminology.

Why the Pairing Must Be Exact

The summed column row pair requires that the number of columns in the first matrix exactly equal the number of rows in the second matrix, since each value of the shared label must correspond to an actual entry present in both factors for the sum defining each product entry to be well formed.

Formal Description

Matching Column Count to Row Count

If the first factor has $m$ columns and the second factor has $n$ rows, the summed column row pair exists only when:

m = n

allowing the column label of the first factor and the row label of the second factor to be identified as a single summed index ranging from $1$ to $m$ .

Expressing the Product Using the Pair

Denoting the shared column-row label by $j$ , the entries of the product tensor $P_{k}^{i}$ are computed by summing over this pair:

P_{k}^{i} = \sum_{j = 1}^{m} A_{j}^{i} B_{k}^{j}

with each term in the sum drawing the $j$ -th column entry of the first factor's row $i$ and the $j$ -th row entry of the second factor's column $k$ .

Visualizing the Pairing

The summed column row pair can be pictured as the moment where the horizontal sequence of entries along a row of the first matrix is laid alongside the vertical sequence of entries along a column of the second matrix, with corresponding positions in each sequence multiplied together before the products are added.

Properties

Absence From the Final Result

Once the summation over the column row pair is complete, this shared label no longer appears in the resulting tensor, leaving only the row label of the first factor and the column label of the second factor as the indices of the product.

Dependence on Consistent Labeling

The summed column row pair relies on both factors using a mutually consistent range for the identified label, meaning any relabeling applied to one factor's column index must be mirrored in the other factor's row index for the sum to remain well defined.

Directionality Reflecting Multiplication Order

Because the summed column row pair specifically links the first factor's columns to the second factor's rows, reversing the order of multiplication would instead require pairing the second factor's columns with the first factor's rows, a generally different and possibly invalid pairing unless the dimensions happen to permit it.

Practical Considerations

Verifying Multiplicability

Before attempting to multiply two matrices, confirming that a valid summed column row pair exists, meaning that the column count of the first matches the row count of the second, is the standard preliminary check for whether the multiplication is defined at all.

Relation to Block Matrix Multiplication

When matrices are partitioned into blocks, the summed column row pair concept extends to the block level, requiring that the column partitioning of the first factor align with the row partitioning of the second factor before block-wise multiplication and summation can proceed.

Pedagogical Role

The summed column row pair offers an accessible, visual entry point into the more abstract tensor contraction description of matrix multiplication, serving as the bridge between elementary matrix arithmetic taught with explicit rows and columns and the index-based formalism used in general tensor algebra.