Monte Carlo method for estimating eigenvalues of symmetric matrices

In this paper, we reveal a new relationship between Rayleigh quotients and residual errors associated with symmetric matrices and unit vectors. We show that for a fixed value of the Rayleigh quotient, the Euclidean norm of the corresponding residual vectors admits both upper and lower bounds. These bounds are related to the eigenvalues of the matrix under consideration. Furthermore, we demonstrate how the derived upper bound can be used to construct a simple Monte Carlo algorithm for estimating the minimum and maximum eigenvalues.