A dynamical system that computes eigenvalues and diagonalizes matrices with a real spectrum

The present paper deals with the problem of diagonalizing matrices using a control system of the form A = [U, A], where [U, A] = UA - AU and A, U are real matrices. It is shown that the feedback U = [N, A + A^T] + p[A^T, A], N diagonal, rho > 0 allows to solve the diagonalization problem under the assumption that the to be diagonalized matrix has real spectrum. Moreover, in the case of a complex spectrum, the feedback allows to check if a matrix is stable or to compute all eigenvalues of a matrix or roots of a polynomial.