Geometrical properties of the Frobenius condition number for positive definite matrices (Q947673)

Let the Frobenius inner product \((A, B)_F = \text{tr}(A^{T}B)\) be defined in the space of square real \(n \times n\) matrices. The geometrical properties of the Frobenius condition number of positive definite matrices in such an inner product space are studied with the aim to get a bound for the ratio between the angle that a matrix \(A\) forms with the identity ray, \(\alpha I\), for \(\alpha > 0\), and the angle that \(A^{-1}\) forms with \(\alpha I\). As a result new lower bounds for the condition number of \(A\) which only require the trace of \(A\) and the Frobenius norm of \(A\) are found. A new practical lower bound for the Frobenius condition number \(\kappa_F(A)\) is given by the expression \(\kappa_F(A) \geq \max(n, \frac{\sqrt{n}}{\cos^{2}(A, I)})\) and its accuracy is evaluated in numerical experiments.