Frobenius matrix norm

Let $R$ be a ring with a valuation $|\cdot |$ and let $M(R)$ denote the set of matrices over $R$. The Frobenius norm function or Euclidean matrix norm is the norm function $||\cdot |{|}_F:M(R)\to ℝ$ given by

${||A||}_F=\sqrt{{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{|a_{ij}|}^2},$

where $m$ and $n$ respectively denote the number of rows and columns of $A$. Note $A$ need not be square for this definition. A more concise (though ) definition, in the case that $R=ℝ$ or $ℂ$, is

${||A||}_F=\sqrt{\text{trace}(A^{*}A)},$

where $A^{*}$ denotes the conjugate transpose of $A$.

Some :

• Denote the columns of $A$ by $A_i$. A nice property of the norm is that

  ${||A||}_F^2={||A_1||}_2^2+{||A_2||}_2^2+\mathrm{\cdots }+{||A_n||}_2^2.$

• Let $A$ be a square matrix and let $U$ be a unitary matrix of same size as $A$. Then ${||A||}_F={||U^{\ast }AU||}_F$ where $U^{\ast }$ is the conjugate transpose of $U$.

• If $AB$ is defined, then ${||AB||}_F\le {||A||}_F{||B||}_F$.

Title	Frobenius matrix norm
Canonical name	FrobeniusMatrixNorm
Date of creation	2013-03-22 11:43:25
Last modified on	2013-03-22 11:43:25
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	25
Author	mathcam (2727)
Entry type	Definition
Classification	msc 65F35
Classification	msc 15A60
Classification	msc 18-00
Synonym	Euclidean matrix norm
Synonym	matrix F-norm
Synonym	Hilbert-Schmidt norm
Related topic	MatrixNorm
Related topic	MatrixPnorm
Related topic	VectorNorm
Related topic	VectorPnorm
Related topic	ShursInequality
Related topic	trace
Related topic	transpose
Related topic	Transpose
Related topic	MatrixLogarithm
Related topic	FrobeniusProduct