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

The condition number cφ of a nonsingular matrix A is defined by cφ(A) = φ(A)φ(A1) where ordinarily φ is a norm. It was proved by O. Taussky-Todd that (c) cφ(A) cφ(AA) when φ(A) = (trAA)12 and when φ(A) is the maximum absolute characteristic root of A. It is shown that (c) holds whenever φ is a unitarily invariant norm, i.e., whenever φ satisfies φ(A) > 0 for A0; φ(αA) = |α|φ(A) for complex α; φ(A + B) φ(A) + φ(B); φ(A) = φ(AU) = φ(AU) for all unitary U. If in addition, φ(Eij) = 1, where Eij is the matrix with one in the (i,j)-th place and zeros elsewhere, then cφ(A) [cφ(AA)]12. 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
Received: 10 March 1964
Published: 1 March 1965
Albert W. Marshall
Ingram Olkin