The following two problems are solved: 1. A matrix A 2 Mn(C) is known to be a rank r perturbation (where r n) of a Hermitian matrix. In other words, A can be represented as A = H + R, where H = H and rank(R) = r. However, the matrices H and R are not known. The question is how to find these matrices. 2. A matrix A 2 Mn(C) is known to be a rank r perturbation (where r n) of a unitary matrix; i.e., A = U +R, where UU = In and rank(R) = r. Find U and R.
