14.10.2 Tensor Matrix Product Entry Formula

The Tensor Matrix Product Entry Formula explains how tensor components interact via matrix multiplication in multilinear algebra.

Tensor Matrix Product Entry Formula is the explicit expression that computes any single entry of a tensor product matrix directly from the entries of the two factor matrices, without requiring the full composite matrix to be built first. It gives a closed-form rule connecting a pair of indices in the composite matrix to a corresponding pair of index pairs in the factor matrices.

Statement of the Formula

General Entry Expression

For a first factor matrix with row extent m₁ and column extent n₁, and a second factor matrix with row extent m₂ and column extent n₂, any entry of the composite matrix at row i and column j is the product of one entry from the first factor and one entry from the second factor.

C_{i j} = A_{i_{1} j_{1}} B_{i_{2} j_{2}}

Index Decomposition Rule

The row index i and column index j of the composite matrix decompose into outer and inner components using the row and column extents of the second factor matrix as the step size.

i_{1} = ⌈ i / m_{2} ⌉

i_{2} = (i - 1) mod m_{2} + 1

Worked Illustration

Example With Small Matrices

Given a first factor matrix with two rows and two columns and a second factor matrix with two rows and two columns, the entry of the composite matrix at row three and column three is computed by decomposing the indices and multiplying the corresponding entries.

Consistency With the Block Structure

Entry Formula Within a Single Block

Within any fixed block determined by a single outer index pair, the entry formula reduces to a plain scalar multiple of the second factor matrix, since the outer entry remains constant across the entire block while the inner indices vary over the full range of the second factor.

Entry Formula Across Block Boundaries

Moving from the last entry of one block to the first entry of the next block changes only the outer index, leaving the inner index reset to its starting value, which matches the block-by-block structure of the composite matrix.

Special Forms of the Entry Formula

Diagonal Entries of the Composite Matrix

A composite entry lies on the main diagonal exactly when its row index equals its column index, which happens precisely when both the outer indices are equal to each other and the inner indices are equal to each other.

C_{i i} = A_{i_{1} i_{1}} B_{i_{2} i_{2}}

Zero Entries From Zero Factors

If either the outer entry or the inner entry used in the formula is zero, the resulting composite entry is also zero, so the pattern of zero entries in the composite matrix is fully determined by the patterns of zero entries in the two factor matrices.

Using the Formula Without Full Expansion

Direct Entry Lookup

Because the entry formula depends only on the two decomposed index pairs, any single entry of a very large composite matrix can be obtained by evaluating the formula directly, without ever constructing the full composite matrix in memory.

Sparse Composite Matrices

When either factor matrix has many zero entries, the entry formula shows that the composite matrix inherits sparsity in a structured way, since every composite entry that decomposes to a zero outer or inner entry is guaranteed to vanish.

Extension to More Than Two Factors

Repeated Decomposition

When the composite matrix comes from a tensor product of three or more factor matrices, the entry formula is applied recursively: the composite index is first decomposed against the combined extent of all remaining factors, and the process repeats until each individual factor index has been isolated.

Product of All Factor Entries

The final entry of a composite matrix built from several factor matrices equals the product of one entry taken from each individual factor matrix, with each factor's index obtained from the recursive decomposition of the original composite indices.