Condition Numbers of Random Triangular Matrices

• Authors: D. Viswanath;L. N. Trefethen
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 1998

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 \begin{equation*} \sqrt[n]{\kn} \rightarrow 2 \:\:\: \text{\it almost surely} \end{equation*} as $n\rightarrow\infty$. 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 \begin{equation*} \sqrt[n]{\kn}\rightarrow 1.305683410\ldots \:\:\:\text{\it almost surely} \end{equation*} as $n\rightarrow\infty$. 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.