Let $k$ and $d$ be positive integers such that $d/k:=\lambda > 1$. Let $W$ be $k \times d$ random matrix with rows $w\_1,\ldots,w\_k \in \mathbb R^d$ drawn iid from $N(0,(1/d)I\_d)$, and define the $k \times k$ matrix $C(W)$ by setting $C(W)\_{i,j} := 2(w\_i^\top w\_j)^2 + \|w\_i\|^2\|w\_j\|^2$.



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> **Question.** Is there a high-probability good lower-bound for the smallest eigenvalue of $C(W)$ ?
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**N.B.** I'm familiar with standard RMT.



Lower-bound for smallest eigenvalue of random $k \times $k matrix $C(W)$ defined by $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$
We have
$$ C(W) = 2 A \circ A + v v^\top$$
where $v$ is the vector with entries $\|w\_i\|^2$, $A$ is the Wishart matrix with entries $w\_i^\top w\_j$, and $\circ$ is the Hadamard product. From the [Schur product theorem](https://en.wikipedia.org/wiki/Schur_product_theorem) (and the fact that adding a positive semi-definite matrix to a self-adjoint matrix only serves to increase the least eigenvalue $\lambda\_1$) we conclude that
$$ \lambda\_1(C(W)) \geq 2 \lambda\_1(A)^2.$$
Combining this with the [Marchenko-Pastur law](https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution) one gets an almost sure lower bound of $2 (1 - 1/\sqrt{\lambda})^4 - o(1)$ (if I have not made a sign error). However, the application of the Schur product theorem may be inefficient and one could hope to improve this lower bound slightly.


Note from the Weyl inequalities that
$$ 2\lambda\_1(A \circ A) \leq \lambda\_1(C(W)) \leq 2 \lambda\_2(A \circ A) $$
so the problem really boils down to understanding the spectral behavior of the Hadamard square $A \circ A$ of a Wishart matrix $A$.


