Gramian Matrix Eigenvalues--Stronger Statement than Non-Negative I'm struggling to find conditions under which this holds:
$AA^T - B \succeq 0\,.$
If it helps, A is not necessarily square and $A_{ij} \in \{-1, 0,1\}$. B is diagonal and I would like to find conditions on $B_{ii}$ such that the above statement is true. 
I have googled Gram Matrices but didn't find anything particularly helpful. Any ideas?

If $B=\mu I$ for some $\mu\in\mathbb{R}$, then $AA^T-\mu I\succeq 0$ if and only if $\lambda_{\min}(AA^T)\geq \mu$. 
