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Let $\mathbf{a}_{ij}$, $1\leq i\leq j\leq n$, be independent random variablesand $\mathbf{a}_{ji}=\mathbf{a}_{ij}$, for all $i,j$. Suppose that every$\mathbf{a}_{ij}$ is bounded, has zero mean, and its variance is given by$\sigma_{i}\sigma_{j}$, for a given sequence of positive real numbers$\Psi=\{\sigma_{i} , i\in\mathbb{N}\}$. Hence, the matrix of variances$V_n=\left( \mbox{Var}( \mathbf{a}_{ij}) \right)_{i,j=1}^n$ has rank one forall $n$. We show that the empirical spectral distribution of the symmetricrandom matrix $\mathbf{A}_n(\Psi)$ $ =$ $ \left(\mathbf{a}_{ij}/\sqrt{n}\right)_{i,j=1}^n$ converges weakly (and withprobability one) to a deterministic limiting spectral distribution which wefully characterize by providing closed-form expressions for its limitingspectral moments in terms of the sequence $\Psi$. Furthermore, we propose ahierarchy of semidefinite programs to compute upper and lower bound on theexpected spectral norm of $\mathbf{A}_n$, for both finite $n$ and the limit$n\to\infty$.

Spectral Moments of Random Matrices with a Rank-One Pattern of Variances