A Determinant Maximization Problem Occurring in the Theory of Data Communication

Abstract

Let H be a Hermitian $m \times m$ matrix, with eigenvalues $\lambda _1 \geqq \lambda _2 \geqq \cdots \geqq \lambda _m \geqq 0$ of which $m^ + \geqq 1$ are positive. Then the maximum of $\det ( {I_n + GHG^ * } )$ over all complex $n \times m$ matrices G) satisfying trace $GG^ * \leqq 1$ is \[ r^{ - r} \left( {1 + \mathop \Sigma \limits_{i = 1}^r \lambda _i^{ - 1} } \right)^r \mathop \Pi \limits_{j = 1}^r \lambda _j , \] where r is the largest integer satisfying $r\leqq n,r\leqq m^ + $, and $r< \lambda _r ( {1 + \Sigma _{i = 1}^r \lambda _i^{ - 1} } )$ . The matrices G for which the maximum is attained are characterized. They have rank r. A special case of this problem arises in the optimization of certain data communication systems.