Expected Values of Scalar-Valued Functions of a Complex Wishart Matrix

Abstract

The complex Wishart distribution has ample applications in science and engineering. In this paper, we give explicit expressions for E(tr(Wh))g(tr(Wj))i and E(tr(W−h))g(tr(W−j))i, respectively, for particular values of g, h, i, j, g+h+i+j≤5, where W follows a complex Wishart distribution. For specific values of g, h, i, j, we first write (tr(Wh))g(tr(Wj))i and (tr(W−h))g(tr(W−j))i in terms of zonal polynomials and then by using results on integration evaluate resulting expressions. Several expected values of matrix-valued functions of a complex Wishart matrix have also been derived.

Keywords

• complex matrix
• moments
• multivariate gamma function
• random matrix
• trace
• zonal polynomials