# Lower-bound for smallest eigenvalue of random $k \times$k matrix $C(W)$ defined by $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$

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 $$C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$$.

Question. Is there a high-probability good lower-bound for the smallest eigenvalue of $$C(W)$$ ?

N.B. I'm familiar with standard RMT.

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 (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 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 $$A$$.