# 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 $||\,\cdot\,||_{F}:M(R)\rightarrow\mathbb{R}$ given by

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

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 $\mathbb{C}$, is

 $\displaystyle||\,A\,||_{F}=\sqrt{\textrm{trace}(A^{*}A)},$

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

 $\displaystyle||A||_{F}^{2}=||A_{1}||_{2}^{2}+||A_{2}||_{2}^{2}+\cdots+||A_{n}|% |_{2}^{2}.$
• 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 $||\,AB\,||_{F}\leq||\,A\,||_{F}\ ||\,B\,||_{F}$.

 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