On the structural eigenvalues of block random matrices
The paper concerns the eigenvalues of an \(n\times n\) random matrix \(A_n=(a_{ij})\) consisting of \(m^2\) square blocks. The entries are such that \(a_{ij}-\mathbb E a_{ij}\) are independent identically distributed random variables with variance \(\sigma^2\) while the averages \(\mathbb E a_{ij}\) differ from one block to another and are equal for all entries lying inside a block. The main result is that in the limit of infinite \(n\), \(A_n\) has \(m\) eigenvalues that are of order \(n\); the magnitude of these is determined by the values of \(\mathbb E a_{ij}\). The rest eigenvalues are of order \(\sqrt n\) with magnitude proportional to \(\sigma\). The proof is based on a combination of perturbation theorems for eigenvalues of matrices with diagonal Jordan normal form and the fundamental fact of random matrix spectral theory that for \(\widehat A_n=A_n-\mathbb E A_n\) one has \[ |\widehat A_n|= 2\sigma\sqrt n (1+o(1)),\quad n\to\infty \] [see \textit{Z. D. Bai} and \textit{Y. Q. Yin}, Probab. Theory Relat. Fields 73, 555--569 (1986; Zbl 0586.60021)].
scientific article; zbMATH DE number 951801