Eigenvalues and Condition Numbers of Complex Random Matrices

Eigenvalues and Condition Numbers of Complex Random Matrices

In this paper, the distributions of the largest and smallest eigenvalues of complex Wishart matrices and the condition number of complex Gaussian random matrices are derived. These distributions are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. Several results are derived on complex hypergeometric functions and...

Condition numbers of random matrices

Condition Numbers of Gaussian Random Matrices

Abstract. Let Gm×n be an m × n real random matrix whose elements are independent and identically distributed standard normal random variables, and let κ2(Gm×n) be the 2-norm condition number of Gm×n. We prove that, for any m ≥ 2, n ≥ 2, and x ≥ |n − m| + 1, κ2(Gm×n) satisfies 1 √ 2π (c/x)|n−m|+1 < P ( κ2(Gm×n) n/(|n−m|+1) > x) < 1 √ 2π (C/x)|n−m|+1, where 0.245 ≤ c ≤ 2.000 and 5.013 ≤ C ≤ 6.414...

Condition Numbers of Random Triangular Matrices

Let L n be a lower triangular matrix of dimension n each of whose nonzero entries is an independent N(0; 1) variable, i.e., a random normal variable of mean 0 and variance 1. It is shown that n , the 2-norm condition number of L n , satisses n p n ! 2 almost surely as n ! 1. This exponential growth of n with n is in striking contrast to the linear growth of the condition numbers of random dense...

Eigenvalues of Graded Matrices and the Condition Numbers of a Multiple Eigenvalue Eigenvalues of Graded Matrices and the Condition Numbers of a Multiple Eigenvalue

abstract This paper concerns two closely related topics: the behavior of the eigenvalues of graded matrices and the perturbation of a nondefective multiple eigenvalue. We will show that the eigenvalues of a graded matrix tend to share the graded structure of the matrix and give precise conditions insuring that this tendency is realized. These results are then applied to show that the secants of...