condition numbers of random triangular L.N.T...pdfVIP

Condition Numb ers of Random Triangular Matrices D Viswanath and LN Trefethen y July Abstract Let L b e a lower triangular matrix of dimension n each of whose n nonzero entries is an indep endent N variable ie a random normal variable of mean and variance It is shown that the norm n condition numb er of L satises n p n almost surely n as n This exp onential growth of with n is in striking contrast to n the linear growth of the condition numb ers of random dense matrices with n that is already known This phenomenon is not due to small entries on the diagonal ie small eigenvalues of L Indeed it is shown that a n lower triangular matrix of dimension n whose diagonal entries are xed at with the sub diagonal entries taken as indep endent N variables is also exp onentially illconditioned with the norm condition numb er n of such a matrix satisfying p n almost surely n as n A similar pair of results ab out complex random triangular matrices is established The results for real triangular matrices are gen eralized to triangular matrices with entries from any symmetric strictly stable distribution Intro duction Random dense matrices are wellconditioned Edelman has shown that if each of the n entries o