Norms and inequalities for condition numbers

Albert W. Marshall and Ingram Olkin

Vol. 15 (1965), No. 1, 241–247

DOI: 10.2140/pjm.1965.15.241

Abstract

The condition number $c_{\phi }$ of a nonsingular matrix $A$ is defined by $c_{\phi }\left(A\right)=\phi \left(A\right)\phi \left(A^{-1}\right)$ where ordinarily $\phi$ is a norm. It was proved by O. Taussky-Todd that $\left(c\right)$ $c_{\phi }\left(A\right)\leqq c_{\phi }\left(A{A}^{\ast }\right)$ when $\phi \left(A\right)={\left(trA{A}^{\ast }\right)}^{1∕2}$ and when $\phi \left(A\right)$ is the maximum absolute characteristic root of $A$. It is shown that $\left(c\right)$ holds whenever $\phi$ is a unitarily invariant norm, i.e., whenever $\phi$ satisfies $\phi \left(A\right)>0$ for $A\ne 0$; $\phi \left(\alpha A\right)=\left|\alpha \right|\phi \left(A\right)$ for complex $\alpha$; $\phi \left(A+B\right)\leqq \phi \left(A\right)+\phi \left(B\right)$; $\phi \left(A\right)=\phi \left(AU\right)=\phi \left(AU\right)$ for all unitary $U$. If in addition, $\phi \left(E_{ij}\right)=1$, where $E_{ij}$ is the matrix with one in the $\left(i,j\right)$-th place and zeros elsewhere, then $c_{\phi }\left(A\right)\geqq {\left[c_{\phi }\left(A{A}^{\ast }\right)\right]}^{1∕2}$. Generalizations are obtained by exploiting the relation between unitarily invariant norms and symmetric gauge functions. However, it is shown that (c) is independent of the usual norm axioms.

Mathematical Subject Classification

Primary: 15.58

Milestones

Received: 10 March 1964

Published: 1 March 1965