4
$\begingroup$

Given 2 (symmetric) PSD matrices $A,B$, is the following set $S_{A,B}$ non-empty?

$$ S_{A,B} = \{ C: C\succeq A, C\succeq B, \text{ and }\forall D, D\succeq A, D\succeq B \implies D\succeq C \} $$

If $S_{A,B}$ is non-empty, it has to be unique due the Loewner order being antisymmetric. Such an element would, in some sense, would constitute a PSD maximum of $A,B$.

If $S_{A,B}$ is non-empty, is there a poly-time algorithm to compute it?

Improve this question
$\endgroup$
1
  • $\begingroup$ Wondering how would this work for matrices $A$ and $B$ which commute. In that case, one can do a simultaneous diagonalization. Say $A=UD_AU^H$ and $B=UD_BU^H$. Then would $C = U \max\{D_A, D_B\} U^H$ work? $\endgroup$
    – DSM
    Commented Aug 4, 2020 at 5:18

1 Answer 1

Reset to default
3
$\begingroup$

Kadison in this paper proves that self-adjoint $A$ and $B$ will have a greatest lower bound if and only if they are comparable (meaning $A\leq B$ or $B\leq A$ or $A=B$). Another way to put this is that the Loewner order is an anti-lattice.

Since this is for all self-adjoint then one can flip the order by multiplying by $-1$ to get that $A$ and $B$ will have a least upper bound if and only if they are comparable.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .