Polynomial Expression for Distribution of the Smallest Eigenvalue of Wishart Matrices

Wishart matrix and its eigenvalues are very important to the analysis of multiple-antenna systems. In this paper, we derive a polynomial expression for the distribution of the smallest eigenvalue of Wishart matrices. Compared with earlier results in matrix determinant forms, our expression is much easier to handle. The expression obtained applies for Wishart matrices of arbitrary orders and is an extension to Edelman´s and Park´s results. Moreover, our work here can also be viewed as an interesting proof of one complicated equality, since the distribution is unique and our result must be equivalent to earlier researchers´. In a word, with the result of this paper, further mathematical analysis on the statistics of the smallest eigenvalue becomes possible.