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Let H be a Hermitian m m matrix, with eigenvalues ₁ ₂ ₘ 0 of which m^ + 1 are positive. Then the maximum of (Iₙ + GHG^ *) over all complex n m matrices G) satisfying trace GG^ * 1 is \ r^ - r (1 + ₈ = ₁ʳ ᵢ^{ - 1 }) ʳ ₉ = ₁ʳ ⱼ, \ where r is the largest integer satisfying r n, r m^ +, and r< ᵣ (1 + ₈ = ₁ʳ 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.
H. S. Witsenhausen (Sat,) studied this question.