Function Repository Resource:

# LogarithmicNorm

Evaluate the logarithmic norm of a square matrix

Contributed by: Jan Mangaldan
 ResourceFunction["LogarithmicNorm"][m,p] gives the logarithmic p‐norm of the matrix m.

## Details

m must be a square matrix.
p can be any of 1, 2 or .
The logarithmic norm μp(m) is defined as , where id is the identity matrix of appropriate size and the norm satisfies .

## Examples

### Basic Examples (3)

Logarithmic 1-norm of a 3×3 matrix:

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Logarithmic -norm of a 3×3 matrix:

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Logarithmic 2-norm of a 3×3 matrix:

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### Scope (2)

A 3×3 matrix:

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Evaluate the logarithmic norm with exact arithmetic:

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Evaluate the logarithmic norm with machine arithmetic:

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Evaluate the logarithmic norm with 20­digit arbitrary precision arithmetic:

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Logarithmic norm of a sparse matrix:

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### Properties and Relations (2)

The logarithmic norm can be negative, and is thus not a matrix norm:

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The logarithmic 2-norm of m is equal to the largest eigenvalue of :

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## Version History

• 1.0.0 – 19 January 2022