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 matrices with n that is already known. This phenomenon is not due to small entries on the diagonal (i.e., small eigenvalues) of L n. Indeed, it is shown that a lower triangular matrix of dimension n whose diagonal entries are xed at 1 with the subdiagonal entries taken as independent N(0; 1) variables is also exponentially ill-conditioned with the 2-norm condition number n of such a matrix satisfying n p n ! 1:305683410: : : almost surely as n ! 1. A similar pair of results about complex random triangular matrices is established. The results for real triangular matrices are generalized to triangular matrices with entries from any symmetric, strictly stable distribution.