14.17.2 Tensor Map Product Factor Component Multiplication

Tensor Map Product Factor Component Multiplication is the rule stating that any single entry of the matrix representing a combined operator, built from a tensor product of maps, equals the product of one entry taken from the matrix of the first factor map and one entry taken from the matrix of the second factor map, with the specific pair of entries determined by decomposing the composite row and column indices.

Statement of the Rule

The Multiplicative Entry Formula

For a combined operator built from a map on the first factor with matrix entries denoted by a and a map on the second factor with matrix entries denoted by b, the entry of the composite matrix at composite row i and composite column j equals the product of the corresponding entry of a and the corresponding entry of b.

Index Decomposition Feeding the Formula

The outer indices i₁ and j₁, together with the inner indices i₂ and j₂, are obtained from the composite indices i and j exactly as in the general Kronecker product entry formula, using the row and column extents of the second factor's matrix as the decomposition step.

Origin of the Formula Within Map Evaluation

Connection to Basis Vector Images

The entries a and b arise from expressing the images of basis vectors under the first and second factor maps, respectively, in terms of the basis of their own codomain; the factor component multiplication rule shows how these two separately obtained sets of coefficients combine into the coefficients describing the image of a basis elementary tensor under the combined operator.

Consistency With the Elementary Evaluation Rule

Because the combined operator acts on a basis elementary tensor by applying each factor map to its own basis vector independently, the coefficient of any resulting basis elementary tensor in the codomain is necessarily the product of the two individual coefficients, one from each factor map's own expansion.

Diagram of Factor Component Multiplication

One Entry From Each Factor Producing One Composite Entry

The diagram below shows a single entry from the first factor's matrix and a single entry from the second factor's matrix combining by multiplication to produce a single entry of the composite matrix.

Consequence for Zero and Nonzero Entries

Zero Entries From Either Factor

If either the outer entry a or the inner entry b is zero for a given pair of indices, the corresponding composite entry is also zero, since multiplying by zero in either factor immediately produces zero regardless of the value of the other factor.

Nonzero Entries Require Both Factors Nonzero

A composite entry is nonzero exactly when both the corresponding outer entry and the corresponding inner entry are individually nonzero, giving a precise characterization of the nonzero pattern of the composite matrix purely in terms of the nonzero patterns of the two factor matrices.

Numerical Stability Considerations

Products of Small or Large Entries

Because a composite entry is a product of two individual entries, an outer entry that is very small combined with an inner entry that is very large, or vice versa, can produce a composite entry of moderate size, while extreme values in both factors simultaneously can lead to very small or very large composite entries.

Error Propagation Through Multiplication

Any numerical error present in either the outer entry or the inner entry propagates multiplicatively into the composite entry, so the accuracy of a computed composite entry depends directly on the accuracy of both contributing factor entries.

Using the Rule Without Full Matrix Construction

Direct Lookup of a Single Composite Entry

Because factor component multiplication depends only on two individual entries selected from the factor matrices, any single entry of a very large composite matrix can be obtained directly, without constructing the full composite matrix, by looking up the appropriate entry in each factor matrix and multiplying them.

Use in Sparse Computation

When either factor matrix has a large number of zero entries, factor component multiplication allows a computation to skip entirely over composite entries known in advance to be zero, since such entries can be identified directly from the nonzero patterns of the two factor matrices without any further calculation.

Extension to Several Factors

Multiplying Entries Across Many Factors

When the combined operator is built from three or more factor maps, factor component multiplication extends by taking the product of one matrix entry from every individual factor, with each factor's entry selected according to the recursive index decomposition used for composite matrices with more than two factors.

Zero Propagation Across Many Factors

With several factors involved, a composite entry is zero as soon as any single one of the contributing factor entries is zero, so the nonzero pattern of a many-factor composite matrix is determined by requiring every one of the individual factor entries, across all factors, to be simultaneously nonzero.