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1991•Linear and Multilinear AlgebraRequires access

Upper bounds for nearly optimal diagonal scaling of matrices

Alexander Shapiro

Open publisher page 6 citations

Abstract

Optimal diagonal scaling of an n×n matrix A consists in finding a diagonal matrix D that minimizes a condition number of AD. Often a nearly optimal scaling of A is achieved by taking a diagonal matrix D 1 such that all diagonal elements of D 1 AT AD 1 are equal to one. It is shown in this paper that the condition number of AD 1 can be at least (n/2)1/2 times the minimal one. Some questions for a further research are posed.

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What this paper is about

Optimal diagonal scaling of an n×n matrix A consists in finding a diagonal matrix D that minimizes a condition number of AD. Often a nearly optimal scaling of A is achieved by taking a diagonal matrix D 1 such that all diagonal elements of D 1 AT AD 1 are equal to one. It is shown in this paper that the condition number of AD 1 can be at least (n/2)1/2 times the minimal one. Some questions for a further research are posed.

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OpenAlex reports 6 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

Optimal diagonal scaling of an n×n matrix A consists in finding a diagonal matrix D that minimizes a condition number of AD. Often a nearly optimal scaling of A is achieved by taking a diagonal matrix D 1 such that all diagonal elements of D 1 AT AD 1 are equal to one. It is shown in this paper that the condition number of AD 1 can be at least (n/2)1/2 times the minimal one. Some questions for a further research are posed.

Key concepts: Mathematics, Diagonal, Scaling, Diagonal matrix, Combinatorics, Matrix (chemical analysis), Geometry, Materials science

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