Condition Numbers of Random Triangular Matrices

Let Ln 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 kn, the 2-norm condition number of Ln , satisfies {equation*} √[n]{} → 2 \:\:\: almost surely {equation*} as n→∞. This exponential growth of kn 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 Ln . Indeed, it is shown that a lower triangular matrix of dimension n whose diagonal entries are fixed at 1 with the subdiagonal entries taken as independent N(0,1) variables is also exponentially ill conditioned with the 2-norm condition number kn of such a matrix satisfying {equation*} √[n]{}→ 1.305683410… \:\:\: almost surely {equation*} as n→∞. 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.