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This question is about the positive definiteness of a (non-random) matrix that is defined using random variables as follows: We fix the vector $v=(1,1)$ (yet, it seems the final result does not ...

Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$. I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...