Suppose that $A,B$ are non-zero, positive semidefinite, $n \times n$ real matrices and that $AB=0$. What can we say about $A$ and $B$?
More precisely, given some $A\neq 0$ non-zero, positive semidefinite, $n \times n$ real matrix. What are the minimal set of conditions on a non-zero, positive semidefinite, $n \times n$ real matrix $B$ to guarantee that $AB \neq0$?
Thanks for any help.
EDIT: $A,B$ are symmetric.
Best Answer
If $A,B\neq0$ then $AB=0$ if and only if $A\bot B$. This also means that the vectors of $A,B$ each span some subspace in $\mathbb{R}^n$, they span a disjoint space, and that $Span\{A\}\cup Span\{B\} = \mathbb{R}^n$. We also conclude that the dimension of the subspaces $dim_A+dim_B = n$.
Therefore if you have a given matrix $A$, it's sufficient to set $B$ such that $Span\{A\}\cap Span\{B\}\neq \emptyset$.