This book gives a coherent and detailed description of analytical methods devised to study random matrices. Given the distribution of matrix elements satisfying certain symmetry conditions, the problem is to find the distribution of quantities depending on a few of its eigenvalues. The passage from matrix elements to all the eigenvalues is simpler than that from all the eigenvalues to a few of them. To achieve this purpose one introduces two kinds of skew-orthogonal polynomials and the method of integration over alternate variables. In the limit of large matrices one is led to the theory of integral equations and non-linear differential equations. All this is relevent to describe nuclear excitations, ultra-sonic resonances of structural materials, spectra of chaotic systems, zeros of Riemann and other zeta functions and in general, the characteristic energies of any sufficiently complicated system. The same mathematiical tools can be hopefully applied in the study of stationary random processes. Since the publication of the second editiion of Random Matrices in 1991, an old result has been better appreciated and many new ones have emerged.
This revised and enlarged edition reflects these developements. For example, the theory of skew-orthogoanl and bi-orthogonal polynomials, parallel to that of the widely known and used orthogonal polynomials, is explained here for the first time. As the new material added one may list the intimate relations among the three classic ensembles (orthogonal, unitary and symplectic), power series expansions of the spacing functions, use fo non-linear differential equations to deduce power series and asymptotic expansions surpassing the previously used inverse scattering method, statistical properties of Gaussian real matrices without symmetry, correlations for Hermitian matrices coupled in a chain, probability density of the determinants of matrices taken from various matrix ensembles, and the relatiion between random permutations to the so called unitary ensembles, circular or Gaussian.