Upper and Lower Bounds for the Tails of the Distribution of the Condition Number of a Gaussian Matrix

Let A be an m × m real random matrix with independently and identically distributed standard Gaussian entries. We prove that there exist universal positive constants c and C such that the tail of the probability distribution of the condition number κ (A) satisfies the inequalities c/x<\κ (A)>m x\<C/x for every $x>1$. The proof requires a new estimation of the joint density of the largest and the smallest eigenvalues of A T A which follows from a formula for the expectation of the number of zeros of a certain random field defined on a smooth manifold.