On the condensed density of the generalized eigenvalues of pencils of Gaussian random matrices and applications
Let \({\mathbf G}\) be a \(p \times p\) random matrix with eigenvalues \(\{\xi_j, j = 1, \dots, p \}\) which form a set of exchangeable random variables. The marginal density \(h(z), ~z \in {\mathbb C}\), of these eigenvalues is called condensed density and represents the expected value of the random normalized counting measure on zeros of \({\mathbf G}\), i.~e., NEWLINE\[NEWLINEh(z) = \frac{1}{p}E \left[ \sum_{j=1}^{p} \delta(z - \xi_j)\right]. NEWLINE\]NEWLINE If a pencil \({\mathbf G} = ({\mathbf G}_1, {\mathbf G}_0)\) of random matrices is considered, the condensed density of its generalized eigenvalues can be computed using the logarithmic potential given by \(\frac{1}{p}E\{\text{log}(|\text{det}({\mathbf G}_1 - z {\mathbf G}_0)|^2) \}\).NEWLINENEWLINE In this paper, a method to estimate the condensed density of the generalized eigenvalues of a pencil \({\mathbf G} = ({\mathbf G}_1, {\mathbf G}_0)\) of Gaussian random matrices is proposed. Within this method the solution of the complex exponentials approximation problem (CEAP) is used. A nonlinear transformation of the pencil is also considered which allows to compute in closed form the expectation evaluation value to the joint probability measure of the elements of the matrices \(({\mathbf G}_1, {\mathbf G}_0)\). In the CEAP case the computational problems can be reduced and \(h(z)\) can be smoothed to some extent acting on a parameter of the approximant. The case with Hankel matrices is discussed from the computational point of view and numerical examples are given.
scientific article; zbMATH DE number 6071637
scientific article