Eigenvalues and Condition Numbers of Random Matrices

Given a random matrix, what condition number should be expected? This paper presents a proof that for real or complex $n \times n$ matrices with elements from a standard normal distribution, the expected value of the log of the 2-norm condition number is asymptotic to $\log n$ as $n \to \infty$. In fact, it is roughly $\log n + 1.537$ for real matrices and $\log n + 0.982$ for complex matrices as $n \to \infty$. The paper discusses how the distributions of the condition numbers behave for large n for real or complex and square or rectangular matrices. The exact distributions of the condition numbers of $2 \times n$ matrices are also given. Intimately related to this problem is the distribution of the eigenvalues of Wishart matrices. This paper studies in depth the largest and smallest eigenvalues, giving exact distributions in some cases. It also describes the behavior of all the eigenvalues, giving an exact formula for the expected characteristic polynomial.