Kronecker Products and Vectorization

Multiplying two matrices normally means combining rows and columns to produce another matrix of compatible size. The Kronecker product, written $A \otimes B$, does something different: it takes every single entry of $A$ and replaces it with that entry times the entire matrix $B$, stitching the results into one large block matrix. A $2\times2$ matrix $A$ Kronecker-multiplied by a $3\times3$ matrix $B$ produces a $6\times6$ matrix made of four $3\times3$ blocks, each block a scaled copy of $B$.

This construction earns its keep alongside vectorization, the operation that stacks all the columns of a matrix into one long column vector, written $\text{vec}(M)$. Together they let you rewrite an equation like $AXB = C$, which involves an unknown matrix $X$ sandwiched between two known ones, as the plain linear system $(B^\top \otimes A)\,\text{vec}(X) = \text{vec}(C)$. That form can be solved with ordinary matrix inversion, whereas the original sandwiched form cannot.

This machinery shows up whenever a model has a natural two-dimensional structure — for instance covariance across both assets and time factors, or the coefficients of a multivariate regression — because it lets those richer models be estimated with the same linear-algebra routines built for ordinary vectors.