Curves on $S^{n - 1} $ That Lead to Eigenvalues or Their Means of a Matrix

Abstract

This paper discusses two dynamical systems on the unit sphere $S^{n - 1} $ in $\mathbb{R}^n $ space, each defined in terms of a real square matrix M. The solutions of these systems are found to converge to points which provide essential information about eigenvalues of the matrix M. It is shown, in particular, how the dynamics of the second flow is analogous to that of the Rayleigh quotient iterations.