Condition Numbers of Matrices

denote its Euclidean operator norm (often called the 2-norm). If  is nonsingular, then its condition number () is defined by () = kk°°−1°° = 1() () where 1 ≥ 1 ≥    ≥  ≥ 0 are the singular values of . The s constitute lengths of the semi-axes of the hyperellipsoid  = { : kk = 1} in -dimensional space; thus  measures elongation of  at its extreme [1]. The role that  plays in numerical analysis cannot be overstated: real matrices with large  are called ill-conditioned whereas matrices with small  are called well-conditioned. In a nutshell,  quantifies the sensitivity of  to pertubations in  and  when solving the linear system  = . It remains to understand meaning of “large” versus “small” in this context. Let the entries of  be independent normally distributed random variables with mean 0 and variance 1. Edelman [2] proved that the condition number  satisfies