Minimization problems for certain structured matrices

Abstract

For given $Z,B\in \mathbb{ C}^{n\times k}$, the problem of finding $A\in \mathbb{C}^{n\times n}$, in some prescribed class ${\cal W}$, that minimizes $\|AZ-B\|$ (Frobenius norm) has been considered by different authors for distinct classes ${\cal W}$. Here, we study this minimization problem for two other classes which include the symmetric Hamiltonian, symmetric skew-Hamiltonian, real orthogonal symplectic and unitary conjugate symplectic matrices. We also consider (as others have done for other classes ${\cal W}$) the problem of minimizing $\|A-\tilde{A}\|$ where $\tilde{A}$ is given and $A$ is a solution of the previous problem.

The key idea of our contribution is the reduction of each one of the above minimization problems to two independent subproblems in orthogonal subspaces of $\mathbb{C}^{n\times n}$. This is possible due to the special structures under consideration. We have developed MATLAB codes and present the numerical results of some tests.