Geometrical properties of the Frobenius condition number for positive definite matrices

We study the geometrical properties of the Frobenius condition number on the cone of symmetric and positive definite matrices. This number, related to the cosine of the angle between a given matrix and its inverse, is equivalent to the classical 2-norm condition number, but has a direct and natural geometrical interpretation. In particular we establish bounds for the ratio between the angle that a matrix forms with the identity ray and the angle that the inverse of that matrix forms with the identity ray. These bounds allow us to establish new lower bounds for the condition number, that only require the trace and the Frobenius norm of the matrix.