A matrix lower bound
- LBNL Library
A matrix lower bound is defined that generalizes ideas apparently due to S. Banach and J. von Neumann. The matrix lower bound has a natural interpretation in functional analysis, and it satisfies many of the properties that von Neumann stated for it in a restricted case. Applications for the matrix lower bound are demonstrated in several areas. In linear algebra, the matrix lower bound of a full rank matrix equals the distance to the set of rank-deficient matrices. In numerical analysis, the ratio of the matrix norm to the matrix lower bound is a condition number for all consistent systems of linear equations. In optimization theory, the matrix lower bound suggests an identity for a class of min-max problems. In real analysis, a recursive construction that depends on the matrix lower bound shows that the level sets of continuously differential functions lie asymptotically near those of their tangents.
- Research Organization:
- Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, CA (US)
- Sponsoring Organization:
- USDOE Director, Office of Science. Office of Advanced Scientific Computing Research. Mathematical, Information, and Computational Sciences Division (US)
- DOE Contract Number:
- AC03-76SF00098
- OSTI ID:
- 836372
- Report Number(s):
- LBNL--50635
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
ALGEBRA
CONDITION NUMBER DISTANCE TO RANK DEFICIENCY FUNCTIONAL ANALYSIS IN MATRIX THEORY IMPLICIT FUNCTION THEOREM LEVEL SETS MATRIX INEQUALITIES MATRIX LOWER BOUND MIN-MAX PROBLEMS TRIANGLE INEQUALITY
CONSTRUCTION
FUNCTIONAL ANALYSIS
MATRICES
NUMERICAL ANALYSIS
OPTIMIZATION