Lower bounds for the spectral radius of a matrix

We develop lower bounds for the spectral radius of symmetric, skew-symmetric, and arbitrary real matrices. Our approach utilizes the well-known Leverrier-Faddeev algorithm for calculating the coefficients of the characteristic polynomial of a matrix in conjunction with a theorem by Lucas which states that the critical points of a polynomial lie within the convex hull of its roots. Our results generalize and simplify a proof recently published by Tarazaga for a lower bound on the spectral radius of a symmetric positive definite matrix. In addition, we provide new lower bounds for the spectral radius of skew-symmetric matrices. We apply these results to a problem involving the stability of fixed points in recurrent neural networks.