Predicting maximum and minimum eigenvalues of a random matrix : A study in simulation for Poisson distribution

Abstract

As we know that, collection of data from large population is very difficult, so in this paper we choose the method of simulation to generate random matrices of order 10×10 whose elements are from Poisson distribution with parameter λ. In the first study, for different values of λ, we generate 100 matrices of order 10×10 and obtain the mean of maximum eigenvalue of each of the 100 matrices using MATLAB and plot a graph between mean of maximum eigenvalues and parameter λ. Finally, we obtain the best curve fit. The equation of best fit enables us to predict the maximum eigenvalue of a random matrix for a given λ. The same procedure is followed for minimum eigenvalue case. In the second study, we repeat the process for random matrices of order 5×5 and observe that the regression equation for predicting the maximum or minimum eigenvalue does not get affected by reducing the order of the matrix. The paper also includes a theoretical analysis of predicting the range of the sum of all the eigenvalues of a diagonalizable random matrix with the help of its trace and Chebyshev’s inequality.