Optimal scaling for p‐norms and componentwise distance to singularity

We give lower and upper bounds for the optimal p‐norm condition number achievable by two‐sided diagonal scalings. There are no assumptions on the irreducibility of certain matrices. The bounds are shown to be optimal for the 2‐norm. For the 1‐norm and inf‐norm the (known) exact value of the optimal condition number is confirmed. We give means how to calculate the minimizing diagonal matrices. Furthermore, a class of new lower bounds for the componentwise distance to the nearest singular matrix is given. They are shown to be superior to existing ones.