Let $A\in\mathbb R^{n\times n}$, $B\in\mathbb R^{m\times m}$ be symmetric, positive definite, matrices. Let $C = A\otimes B$ be their tensor product. I want to compute the Cholesky decomposition of $C$. Suppose I have available the Cholesky decompositions of $A$ and $B$. Is there a way to exploit this information in computing the decomposition of $C$?
Cholesky decomposition of tensor product
cholesky decompositionkronecker productlinear algebramatrix decompositiontensor-products
Best Answer
A Cholesky decomposition of a matrix $M$ is a factorization of form $M=LL^T$, where $L$ is lower triangular and $T$ denotes transpose.
So given Cholesky decompositions $A=\alpha\alpha^T$ and $B=\beta\beta^T$ one can obtain the factorization $A\otimes B = (\alpha \otimes \beta)(\alpha^T\otimes\beta^T)$. To see that this actually is a Cholesky decomposition one checks that $L=\alpha \otimes \beta$ is lower triangular. Inspection of examples should make this clear, but the only formal proof I can think of is notationally complex, involving expressions relating row numbers of $L$ to those of $\alpha$ and $\beta$ with formulas like $k=mi+j$ and statements like $mi+j>mr+s$ implies at least one of $i>r$ and $j>s$ holds.