I have two matrices $A \in \mathcal{M}_{n,d}(\mathbb{R})$ and $B \in \mathcal{M}_{d,d}(\mathbb{R})$ with $B$ being symmetric definite-positive.
I am trying to find a condition on $A$ for which I have : $$Tr(B)Tr(AB^{-1}B^{-1}A^T) < 1$$ or, said differently, by writing $B = M^TM$ with $M \in \mathcal{M}_{n,d}(\mathbb{R})$ : $$\|(M^TM)^{-1}A^T\|\|M\| < 1$$ For the moment, all I have done is write this with coordinates and try to consider condition numbers but I am clearly lost. This specific problem arised when I was working on perturbating a least squares problem, that is why I am considering the condition numbers. I have no other information on the matrices but it is ok to specify conditions for which the inequalities above work.
In any case, $$ \|A\|\le \frac 1{ \|B^{-1}\|\|M\| } $$ is enough.