Eigenvalues, eigenvectors, and random-matrix theory

Abstract

We study methods for calculating eigenvector statistics of random matrix ensembles, and apply one of these methods to calculate eigenvector components of Toeplitz ± Hankel matrices. Random matrix theory is a broad field with applications in heavy nuclei scattering, disordered metals and quantum billiards. We study eigenvalue distribution functions of random matrix ensembles, such as the n-point correlation function and level spacings. For critical systems, with eigenvalue statistics between Poisson and Wigner-Dyson, the eigenvectors can have multifractal properties. We explore methods for calculating eigenvector component expectation values. We apply one of these methods, referred to as the eigenvector-eigenvalue identity, to show that the absolute values of eigenvector components of certain Toeplitz and Toeplitz±Hankel matrices are equal in the limit of large system sizes.