The continuous-time Rayleigh quotient flow on the sphere

A continuous-time differential equation inspired by the Rayleigh quotient iteration for a symmetric matrix A is studied. The set of all continuous solutions, termed the Rayleigh quotient flow, is shown to be a time-scaled version of the Newton flow for Rayleigh quotient cost functional. The scaling factor ensures that the rate of variation of the Rayleigh quotient is constant and positive along solutions. This interpretation leads to a precise phase portrait for the Rayleigh quotient flow. It is shown that a complete (non-degenerate) solution of the Rayleigh quotient flow visits each of the eigenvectors of A in ascending order.