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 $\cdot {}_{F}:M(R)\to \mathbb{R}$ 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 $R=\mathbb{R}$ or $\u2102$, is
${A}_{F}=\sqrt{\text{trace}({A}^{*}A)},$ 
where ${A}^{*}$ denotes the conjugate transpose^{} of $A$.
Some :

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Denote the columns of $A$ by ${A}_{i}$. A nice property of the norm is that
${A}_{F}^{2}={{A}_{1}}_{2}^{2}+{{A}_{2}}_{2}^{2}+\mathrm{\cdots}+{{A}_{n}}_{2}^{2}.$ 
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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 $U$.

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If $AB$ is defined, then ${AB}_{F}\le {A}_{F}{B}_{F}$.
Title  Frobenius matrix norm 
Canonical name  FrobeniusMatrixNorm 
Date of creation  20130322 11:43:25 
Last modified on  20130322 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 1800 
Synonym  Euclidean matrix norm 
Synonym  matrix Fnorm 
Synonym  HilbertSchmidt 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 