On the distribution of a scaled condition number

In this note, we give the exact distribution of a scaled condition number used by Demmel to model the probability that matrix inversion is difficult. Specifically, consider a random matrix A and the scaled condition number κ D ( A ) = ‖ A ‖ F ∙ ‖ A − 1 ‖ {\kappa _D}(A) = {\left \| A \right \|_F} \bullet \left \| {{A^{ - 1}}} \right \| . Demmel provided bounds for the condition number distribution when A has real or complex normally distributed elements. Here, we give the exact formula.