A matrix lower bound

Technical Report · Mon Feb 04 04:00:00 EST 2002

DOI:https://doi.org/10.2172/836372· OSTI ID:836372

Grcar, Joseph F ^[1]

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1. 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

Related Subjects

99 GENERAL AND MISCELLANEOUS
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