A best upper bound for the 2-norm condition number of a matrix Original Research Article

Let A be an n × n nonsingular real or complex matrix. The best possible upper bound for the ratio of the largest and smallest singular values of A, using tr A*A, det A, and n only, is obtained. A comparison with an earlier bound is given, and the singular and nonsquare cases are included. If all the eigenvalues of A are real and positive, the best possible upper bound for the ratio of the largest and smallest eigenvalues of A, involving tr A, det A, and n only, is presented as well.