14.9.1 Tensor Kronecker Matrix Relation

Tensor Kronecker Matrix Relation is the explicit entrywise formula defining the Kronecker product of two matrices, specifying exactly which entry of the larger block matrix corresponds to a given pair of entries drawn from the two original matrices.

The Entrywise Formula

Indexing the Kronecker Product

For a matrix A of size m by n and a matrix B of size p by q, the Kronecker matrix relation defines the entry of A tensor B in block row i and block column j, further indexed within that block by row k and column l, as

giving a single explicit rule mapping the four indices i, j, k, l to one entry of the m p by n q matrix A tensor B.

Reading the Formula as Blocks

The Kronecker matrix relation is equivalently read block by block: the (i, j) block of A tensor B, a p by q submatrix, equals a-i-j times B in its entirety, so the whole matrix is assembled from m n such scaled copies of B, arranged according to the positions of the entries of A.

Consistency of the Two Descriptions

Block Description Implies the Entrywise Formula

Expanding the block description entry by entry recovers exactly the entrywise formula, since the (k, l) entry of the block a-i-j times B is a-i-j times b-k-l, matching the formula stated above at the corresponding position within the larger matrix.

Entrywise Formula Implies the Block Description

Conversely, grouping the entries given by the entrywise formula according to fixed i and j, and letting k and l range over all valid indices, reconstructs exactly the block a-i-j times B, confirming that the two descriptions of the Kronecker matrix relation carry identical information, presented at different levels of granularity.

Size and Structure Consequences

Dimension of the Resulting Matrix

The Kronecker matrix relation produces a matrix of size m p by n q from matrices of size m by n and p by q respectively, so the dimensions multiply rather than add, in direct correspondence with the multiplicative dimension formula for the underlying tensor product spaces.

Sparsity Pattern Inheritance

If A has a zero entry at position (i, j), the entire corresponding block of A tensor B, namely a-i-j times B, is the zero matrix, so any sparsity pattern present in A is inherited by the Kronecker product at the level of whole blocks, a structural fact used in numerical computation to avoid forming or storing blocks known in advance to be zero.

Elementary Examples of the Relation

Kronecker Product with a Scalar Matrix

If A is the one-by-one matrix containing a single scalar c, the Kronecker matrix relation reduces to

recovering ordinary scalar multiplication of a matrix as the special one-dimensional case of the Kronecker matrix relation.

Kronecker Product of Identity Matrices

If A is the m by m identity matrix and B is the p by p identity matrix, the Kronecker matrix relation gives A tensor B equal to the m p by m p identity matrix, since every off-diagonal block a-i-j times B vanishes for i not equal to j, while every diagonal block a-i-i times B equals B itself, namely the identity, placed along the diagonal in exactly the pattern required to reproduce the full identity matrix of the larger size.