How Many Eigenvalues of a Random Matrix Are Real ? Alan

Let A be an n x n matrix whose elements are independent randomvariables with standard normal distributions. As n ..... 00 , the expected numberof real eigenvalues is asymptotic to .J2nln. We obtain a closed form expres-sion for the expected number of real eigenvalues for finite n, and a formula forthe density of a real eigenvalue for finite n. Asymptotically, a real normalizedeigenvalue AI Vn of such a random matrix is uniformly distributed on the in-terval [-I, I] . Analogous, but strikingly different, results are presented for thereal generalized eigenvalues. We report on numerical experiments confirmingthese results and suggesting that the assumption of normality is not importantfor the asymptotic results. DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE,MASSACHUSETTS 02139E-mail address: [email protected]. edu ARTS AND SCIENCES, KAPI'OLANI COMMUNITY COLLEGE, 4303 DIAMOND HEAD ROAD,HONOLULU, HAWAII 98616E-mail address: [email protected] TJ WATSON RSEARCH CENTER, 32-2, IBM, YORKTOWN HEIGHTS, NEW YORK 10598-0218E-mail address:[email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use