Distribution of the smallest eigenvalue of complex central semi-correlated Wishart matrices

The complex central semi-correlated Wishart matrix is the product of a zero-mean row-wise correlated (or columnwise correlated) complex Gaussian matrix and its Hermitian transposition. In this paper, we investigate the distribution of the smallest eigenvalue of the Wishart matrix, which is generated from the square Gaussian matrix. A closed-form expression for distribution of the smallest eigenvalue is derived. Compared with previous results, our expression is very concise, since the distribution has been known in terms of matrix determinants. Numerical results that confirm the expression are also presented. And a complicated equality is proved due to the fact that our distribution is mathematically equivalent to those reported.