Check block ordering in a Kronecker product

Define the task

A Kronecker product replaces each entry of the first matrix with that entry times the complete second matrix. Record which factor describes groups and which describes repeated occasions. Reversing the factors generally changes the arrangement of the resulting blocks, even though the overall dimensions can look plausible.

Carry out the check

Write the intended ordering of the combined index before constructing the matrix. For two two-by-two factors, expect a four-by-four result. Check one complete block and an entry crossing a block boundary. When using a Kronecker product as a covariance model, separately verify the covariance assumptions of both factors and the meaning of separability.

Worked hypothetical example

For hypothetical A with rows (1,2) and (3,4), and B equal to the two-by-two identity, the product has rows (1,0,2,0), (0,1,0,2), (3,0,4,0) and (0,3,0,4). The entry in row one, column three equals two. A reorder of the combined index requires corresponding row and column permutations. A correct numeric product with mislabeled indices can still represent the wrong model.

Method checked 12 November 2026 against R documentation: Kronecker Products on Arrays.

Open the two-by-two kronecker block builder.

Download the kronecker block index worksheet.