# Accession Number:

## AD0677107

# Title:

## NORMS AND INEQUALITIES FOR CONDITION NUMBERS, II,

# Descriptive Note:

# Corporate Author:

## BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB

# Personal Author(s):

# Report Date:

## 1968-09-01

# Pagination or Media Count:

## 13.0

# Abstract:

The condition number c sub phi of a nonsingular matrix A is defined by c sub phi A phi A phi A superscript -1 where ordinarily phi is a norm. It was shown by J. D. Riley that if A is positive definite, c sub phi A kI or c sub phi A when k 0 and phi squared A is the maximum eigenvalue of AA or phi squared A Tr AA. In this paper it is shown more generally that c sub phi A B or c sub phi B when phi satisfies phi U or phi V if V-U is positive definite and when A,B are positive definite satisfying c sub phi A or c sub phi B. Some related inequalities are also obtained. As suggested by Riley, these results may be of practical use in solving a system Ax d of linear equations when A is positive definite but ill-conditioned. Author

# Descriptors:

# Subject Categories:

- Theoretical Mathematics