Minimization problems for certain structured matrices

Abstract

For given Z, B â C^{n\times k}, the problem of finding A â C^{n\times n}, in some prescribed class W, that minimizes ||AZ â B|| (Frobenius norm) has been considered by different authors for distinct classes W. Here, this minimization problem is studied for two other classes, which include the symmetric Hamiltonian, symmetric skew-Hamiltonian, real orthogonal symplectic and unitary conjugate symplectic matrices. The problem of minimizing ||A â AË||, where AË is given and A is a solution of the previous problem, is also considered (as has been done by others, for different classes W). The key idea of this contribution is the reduction of each one of the above minimization problems to two independent subproblems in orthogonal subspaces of C^{n\times n}. This is possible due to the special structures under consideration. Matlab codes are developed, and numerical results of some tests are presented.