Best Available Bounds for Departure from Normality
Steven L. Lee · SIAM Journal on Matrix Analysis and Applications · 1996
The best available bounds for the departure from normality of a matrix are given. The significant properties of these lower and upper bounds are also described. For example, one of the upper bounds is a practical estimate that costs (at most) $2m$ multiplications, where m is the number of nonzeros in the matrix. In terms of applications, the results can be used to bound from above the sensitivity of eigenvalues to matrix perturbations or to bound from below the distance to the closest normal matrix.