Limit Spectra of Random Gram Matrices

Solutions to basic non-linear limit spectral equation for matrices RTR of increasing di-mension are investigated, where R are rectangular random matrices with independent normal entries. The analytical properties of limiting normed trace for the resolvent of RTR are investigated, boundaries of limit spectra found, and upper estimates of spectral density are obtained. Spectra of large random Gram matrices are of interest in “randommatrix physics ” (see [1]), and in solution of systems of empiric linear algebraic equations. First, we cite shortly some recent results stating the convergence of spectral functions obtained earlier. The basic spectral equation (called “canonic equation ” by V.I. Girko [2]) is established that connect limit spectral functions of random Gram matrices with limit spectral function of non-random matrices. Then we suggest an original investigation of solutions of this spectral equation and of the form of limit spectra with full proofs. We consider a sequence P of matrices A and R of size N × n, ξ = R − A, and square matrices Σ = ATA, S = RTR of size n × n as n → ∞, N → ∞, where ξ are random vectors with independent components Denote by Amj, Rmj, ξmj entries o