#### Vol. 15, No. 1, 1965

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Norms and inequalities for condition numbers

### Albert W. Marshall and Ingram Olkin

Vol. 15 (1965), No. 1, 241–247
##### 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 $A\ne 0$; $\phi \left(\alpha A\right)=|\alpha |\phi \left(A\right)$ 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.

Primary: 15.58
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