A best upper bound for the 2-norm condition number of a matrix (Q677922)

The authors derive the best possible upper bound for the ratio of the largest and smallest singular values of an \(n\times n\) nonsingular real or complex matrix \(A\), using \(\text{trace }A^*A\), \(\text{det }A\), and \(n\) only. A comparison with an earlier bound is given, and the singular and nonsquare cases are also included. A best upper bound for the ratio when all the eigenvalues of \(A\) are real and positive is presented as well.