On an eigenflow equation and its structure preserving properties

The present work deals with a dynamical system of the form A¿ = [[N, A^T+A], A]+¿[[A^T, A], A], where A¿ is an n×n real matrix, N is constant n×n real matrix, [A, B] = AB-BA, and ¿ is a positive constant. In particular, the purpose of this work is to establish structure preserving properties of this dynamical system for tridiagonal, Hessenberg, and Hamiltonian matrices in combination with potential applications.