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:

 In:= Out= Logarithmic -norm of a 3×3 matrix:

 In:= Out= Logarithmic 2-norm of a 3×3 matrix:

 In:= Out= ### Scope (2)

A 3×3 matrix:

 In:= Evaluate the logarithmic norm with exact arithmetic:

 In:= Out= Evaluate the logarithmic norm with machine arithmetic:

 In:= Out= Evaluate the logarithmic norm with 20­digit arbitrary precision arithmetic:

 In:= Out= Logarithmic norm of a sparse matrix:

 In:= Out= ### Properties and Relations (2)

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

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

 In:= Out= In:= Out= ## Version History

• 1.0.0 – 19 January 2022