Product of a nilpotent matrix and a diagonal PD matrix

Question

Suppose $A$ is a nilpotent matrix which implies $tr(A)=0$ and $B$ is a PD diagonal matrix. What can be said about $tr(AB)$? Any inequalities regarding this?

Answer 1

There exist no relations between the eigenvalues of $B$ and $tr(AB)$.

Choose, for example, $A=\begin{pmatrix}1&-1\\1&-1\end{pmatrix},B=diag(1,2)$. Then $tr(AB)=-1$.

Now, for every real $t$, $tA$ is nilpotent and $tr((tA)B)=-t$ varies through $\mathbb{R}$ with $t$.