Concise Probability Distributions of Eigenvalues of Real-Valued Wishart Matrices

Abstract: In this paper, we consider the problem of deriving new eigenvalue distributions of real-valued Wishart matrices that arises in many scientific and engineering applications. The distributions are derived using the tools from the theory of skew symmetric matrices. In particular, we relate the multiple integrals of a determinant, which arises while finding the eigenvalue distributions, in terms of the Pfaffian of skew-symmetric matrices. Pfaffians being the square root of skew symmetric matrices are easy to compute than the conventional distributions that involve Zonal polynomials or beta integrals. We show that the plots of the derived distributions are exactly coinciding with the numerically simulated plots.