### Abstract

We prove the strong Elliptical Galactic law for random matrices Ξ n of the general form, i.e. their diagonal entries have nonzero expectations and the pairs of the entries ( , ) have nonzero covariances. In this case the Elliptical Galactic law means that the support of the accompanying spectral density of eigenvalues of matrix Ξ n looks like the picture of several galaxies made by telescope: The picture 1 shows the collision of elliptic supports of the limit spectral density of n.s.f. of random matrix A n + Λ n Ξ n , where A n is a diagonal complex matrix with diagonal entries (0.7, 0), (−1, 0), (0, 0.7i) for corresponding three equal parts of the main diagonal, and random matrix Ξ n has equal covariances ρ (√ ρ = 0.2 + i0.8) of independent pairs of entries ( , ) with zero mean and is multiplied by diagonal matrix Λ n with diagonal entries (1, 0), (0.5, 0.5i), (−1, 0) for corresponding three equal parts of the main diagonal. We have chosen in picture 1 three different diagonal entries of the matrix A n at a short distance. In picture 2, we consider the diagonal matrix A n with diagonal entries (2, 0), (−2, 0), (0, 2i) at a large distant for corresponding three equal parts of the main diagonal. In the letter case we have several domains-supports like ellipses. For the exposition of the Elliptic Law we have chosen the random matrix Ξ n of dimension 30 and 300 its Monte-Carlo simulation. If the distances between the centers of these galaxies are large enough we have several almost elliptical galaxies . These pictures show the elliptic support of the limit spectral density of n.s.f. of random matrix A n + Ξ n , where A n is a diagonal matrix with 5 different diagonal entries (1, 0); (−1, 0); (−0.5,−i); (0, 0.5i); (0, i) and random matrix Ξ n has equal covariances ρ(√ ρ = 0.5 + i0.5)of the entries . We have chosen five different diagonal entries of the matrix A n at a short distance in picture 1 and at a large distant (2, 0); (−2, 0); (−1,−2i); (0, i); (0, 2i) in picture 2. In the letter case we have several domains-supports like ellipses. For the exposition of the Elliptic Law we have chosen the random matrix Ξ n of dimension 50 and 300 its Monte-Carlo simulation. If the distances between the centers of these galaxies are large enough we have several almost elliptical galaxies. Maybe the reader remembers the Monte Carlo simulations of eigenvalues of matrices Ξ n + A n , where Ξ n belongs to the domain of attraction of Circular law and A n is the diagonal matrix whose diagonal entries forms letter R on a complex plain[25]–[27]. For the case when the matrix Ξ n belongs to the domain of attraction of Elliptic law the simulation of eigenvalues of the matrix Ξ n + A n looks like the following picture: These statements are based on the VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Random Independent Arrays . We follow the main strategy of the theory of limit theorems of the probability theory, i.e. we try to solve the problem of description of all limits of normalized spectral functions where λ k ( A n Ξ n B n + C n ) are eigenvalues of non Hermitian matrix A n Ξ n B n + C n , A n , B n , and C n are nonrandom matrices, under general (as only possible) conditions on the entries of random matrices Ξ n , χ is the indicator function. We emphasize that the spectral theory of Hermitian random matrices is rather profound theory [3,13,23,24]. There are essentially three methods of the proof of Elliptic Laws that have been proposed: the REFORM method and Berry-Esseen inequality[11], the method of perpendiculars[15,16], the method of the central limit theorem and limit theorems for eigenvalues of random matrices[23]. The main advantage of REFORM approach is that it enables the results of the previous version of Elliptic law to be extended to the case under consideration. The REFORM-method(or G -martingale approach) enables us to suggest a new method for construction of stochastic canonical equations. In this paper we prove the following Elliptical Galactic Law which generalizes the Strong Circular Law and Weak Circular Law (see the sketch of the proof of this law in the paper V-transform, Dopovidi Akademii nauk Ukrainskoi RSR, Seria A Fizyko-Matematychni ta technichni nauky, 1982, N3, pp.5-6.): For every n , let the pairs of random entries of the complex matrix be independent and given on a common probability space, i , j = 1, ..., n , and where are square complex nonrandom matrices, det A n ≠ 0, det B n ≠ 0, and the real and imaginary parts of random entries have the densities satisfying the corrected Elliptic condition : for some β > 1 or and there exist the densities of the random entries , or the densities of the random entries , satisfying the condition: for some β 1 > 1 the Lyapunov condition is fulfilled: for some δ > 0, Then, with probability one, for almost all x and y where λ k are eigenvalues of the matrix A n Ξ n B n + C n , the Global probability density p n , α ( t , s ) =(∂ 2 /∂ t ∂ s ) F n , α ( t , s ) is equal to where α > 0, where is a block matrix, are entries of the matrix is the block diagonal matrices, whose diagonal block satisfy the system of canonical equations K 97 , j = 1, ..., n , and is a support of the Global probability density, where There exists a unique solution of canonical equation K 97 in the class of positive definite block matrices of the order 2 × 2, analytic in y > 0, t , s .