Matrix Norm

Author

Eric W. Weisstein
April 20, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/LinearAlgebra/MatrixNorm.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MatrixNorm.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<LinearAlgebra`MatrixManipulation`

Usage

?MatrixNorm

Examples

m={{1,2},{3,4}};

ConjugateTranspose[m_List?MatrixQ]:=Transpose[Conjugate[m]]

MatrixNorm[m_List?MatrixQ,1]:=Max[Plus@@@Transpose[Abs[m]]]

MatrixNorm[m_List?MatrixQ,2]:=Sqrt[Max[Eigenvalues[ConjugateTranspose[m].m]]]

MatrixNorm[m_List?MatrixQ,Infinity]:=Max[Plus@@@Abs[m]]

MatrixNorm[m,#]&/@{1,2,∞}

Norm[m,#]&/@{1,2,∞}

Inequality

And@@Less@@@({#[[1]]^2,Times@@Rest[#]}&/@Function[m,Norm[m,#]&/@{2,1,∞}]/@Table[Random[],{10^4},{2},{2}])//Timing

And@@Less@@@({#[[1]],Times@@Rest[#]}&/@Function[m,Norm[m,#]&/@{2,1,∞}]/@Table[Random[],{10},{2},{2}])