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 < P{κ(A) > mx} < C/x for every x > 1. The proof requires a new estimation of the joint density of the largest and the smallest eigenvalues of AT A which follows from a formula for the expectation of the number of zeros of a certain random field defined on a smooth manifold. [ABSTRACT FROM AUTHOR]