1
$\begingroup$

Is there a way to compute $(A\otimes B)x$ quickly without forming the Kronecker product? Often, I'd like to compute the matrix-vector product of a Kronecker product, but I'm not sure of a good way to efficiently produce the product directly. In case it's any easier, I'm also interested in the computing $(A\otimes A)x$.

Thanks for the help!

$\endgroup$
5
$\begingroup$

A basic property of Kronecker products is $$(A\otimes B)\,{\rm vec}(X) = {\rm vec}(BXA^T)$$ where the RHS does not contain any Kronecker products, although it does require a vectorization (and de-vectorization) operation.

Similarly $$(A\otimes A)\,{\rm vec}(X) = {\rm vec}(AXA^T)$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.