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The sum of Wishart matrices has an important role in multiple-input multiple-output (MIMO) multiple access channel (MAC) and MIMO relay channel. In this paper, we present a new closed-form expression for the marginal density of one of the unordered eigenvalues of a sum of K complex central Wishart matrices having covariance matrices proportional to the identity matrix. The expression is general and allows for any set of linear coefficients. The derived expression is used to obtain the ergodic sum-rate capacity for the MIMO-MAC and MIMO relay cases, both as closed-form expressions. We also present a very simple expression to approximate the sum of Wishart matrices by one equivalent Wishart matrix. The agreement between the exact eigenvalue distribution and numerical simulations is perfect, whereas for the approximate solution the difference is indistinguishable.

On the Exact and Approximate Eigenvalue Distribution for Sum of Wishart Matrices