Suppose I have a matrix $A \in \mathcal{M}_{n,n}$ and for fun let's normalize is so that $tr(A) = 1$. Then with the identity matrix $I \in \mathcal{M}_{k,k}$, I'm interested in what happens when we look at

$tr((A \otimes I) (I \otimes A))$

for different $k \leq n$, where we can just think of the tensor as the normal Kronecker product.

For example, if $k = n$, we'd get $tr(A \otimes A) = tr(A)^2 = 1$, and if $k = 1$ we'd just get $tr(A^2)$. I'd like to understand how varying $k$ might move between $[tr(A^2), tr(A)^2]$. Is there some known inequality that addresses this?


Your Answer

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

Browse other questions tagged or ask your own question.