Cholesky decomposition of a Kronecker product

Question

Assume that the $n\times n$ matrix $\mathbf{A}$ has the Cholesky decomposition of the form $\mathbf{A}=\mathbf{L}\mathbf{L}^H$. Now, suppose the matrix $\mathbf{B}$ is the result of a Kronecker product as $\mathbf{B}=\mathbf{I}\otimes\mathbf{I}\otimes\mathbf{A}$ where $\mathbf{I}$ is $2\times 2$ identity matrix. Can we find the Cholesky decomposition of $\mathbf{B}$ in terms of $\mathbf{L}$?

Answer 1

Yes. It is $$ I \otimes I \otimes A = (I \otimes I \otimes L)(I \otimes I \otimes L)^H. $$