 
Summary: UPPER AND LOWER BOUNDS FOR THE TAILS OF THE
DISTRIBUTION OF THE CONDITION NUMBER
OF A GAUSSIAN MATRIX
JEANMARC AZA¨IS AND MARIO WSCHEBOR
SIAM J. MATRIX ANAL. APPL. c 2005 Society for Industrial and Applied Mathematics
Vol. 26, No. 2, pp. 426440
Abstract. 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 (A) satisfies the inequalities
c
x
< P{(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.
Key words. random matrices, condition number, eigenvalue distribution, Rice formulae
AMS subject classifications. 15A12, 60G15
DOI. 10.1137/S0895479803429764
1. Introduction and main result. Let A be an m×m real matrix and denote
