Predicting the Maximum and Minimum Eigenvalues of a Random Matrix Filled with iid Normal Variates: A Study in Simulation
Princila, Soubhik Chakraborty · Journal of Applied Mathematics and Computation · 2021
This paper involves a simulation study to predict the maximum and minimum eigenvalues of a random matrix whose elements are coming from N (µ, σ 2 ).In the first study, we fix σ and for different values of µ we generate 100 matrices of order 10×10 in MATLAB.Then, by plotting a graph between mean maximum eigenvalue and µ, a pattern is detected and we obtain the equation of best curve fit using MS-EXCEL.However, no pattern is detected for mean minimum eigenvalue with respect to μ.In the second study, the same procedure is repeated except that here we fix µ and vary σ.Here the reverse happens interestingly.Pattern is detected for mean minimum eigenvalue with respect to σ but no pattern is detected for the case of mean maximum eigenvalue.Both these studies are repeated for random matrices of order 5×5 with identical results as in the case of 10x10 matrices except that the magnitude of the maximum eigenvalue is reduced by about half when the order of the matrices is reduced by half while magnitude of the minimum eigenvalue is not significantly affected.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.This paper is organised as follows.Section 1 is the introduction.Section 2 is the literature review.Section 3 gives the methodology.Section 4 gives the experimental results and discussion.Section 5 provides some theoretical results for a diagonalizable random square matrix.Finally, Section 6 gives the concluding remarks.