Estimates of the spectral condition number

In this article, new upper and lower bounds for the spectral condition number are obtained. These bounds are constructed based on the Frobenius norm of some matrices related to the given matrix and its inverse. Hence, unlike some existing bounds, these new bounds are smooth functions with respect to the elements in the matrix. It is theoretically established that the new bounds are also sandwiched by the true value of the spectral condition number and its estimates using the Frobenius norms. Moreover, the bounds give the exact value of the spectral condition number when the matrix is unitary or of order less than 3. The new upper bound provided, via statistical numerical comparison, is shown to be the best when compared with existing results.