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

Let A be an $m \times m$ 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 $\kappa (A) $ satisfies the inequalities $\frac{c}{x} m x\} 1$. The proof requires a new estimation of the joint density of the largest and the smallest eigenvalues of ATA which follows from a formula for the expectation of the number of zeros of a certain random field defined on a smooth manifold.

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