What are some lower bounds (if they exist) on $w^t A w$ in terms of $||w||_2^2$ for $A$ an matrix with all positive values? The lower bound can depend on $A$ and $\|w\|_2^2$. $w$ is $k \times 1$. Note when $A = I$ then we exactly get the squared norm of $w$.
If there are no such values, what in the case that $A$ is positive semi-definite? (I.e., $A$ has a $B$ such that $A = BB^T$ and then we are talking $w^tA w = \|Bw\|^2_2$ and we want it to be larger than something that depends on $\|w\|^2_2$).
Thanks.