What are the conditions to make $A \otimes I - B$ positive semi-definite?

Question

I'm curious if there are any clear conditions required of a $2\times 2$ positive semi-definite matrix $A$ (possibly complex) in terms of some $4 \times 4$ matrix $B$ so that $A \otimes I - B$ is positive semi-definite. Here $I$ denotes the $2\times 2$ identity matrix.

Answer 1

Here's something slightly easier than the usual conditions, where $A$ is strictly positive definite, and $B$ is known to be Hermitian:

$$ A \otimes I - B \succeq 0 \iff\\ A \otimes I \succeq B \iff\\ I \succeq [A^{-1/2} \otimes I]B[A^{-1/2} \otimes I] $$ So, your matrix will be positive definite if and only if and only if $[A^{-1/2} \otimes I]B[A^{-1/2} \otimes I]$ has eigenvalues smaller than $1$. This in turn is true if and only if $[A^{-1} \otimes I]B$ has eigenvalues smaller than $1$.

Notably, this requires an inverse, but only a $2 \times 2$ inverse.

Answer 2

$B$ of course must be hermitian, and then all the principal minors of $A \otimes I - B$ must be nonnegative. These are polynomials in the entries of $A$ and $B$ that are not particularly enlightening.