Frobenius matrix normFrobeniusMatrixNorm2001-10-06 14:43:072007-06-24 16:35:15Definition% this is the default PlanetMath preamble. as your knowledge
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\renewcommand{\>}{\rangle}Let $R$ be a ring with a valuation $|\cdot|$ and let $M(R)$ denote the set of matrices over $R$. The \emph{Frobenius norm function} or \emph{Euclidean matrix norm} is the norm function $||\,\cdot\,||_F:M(R)\ra\R$ given by
\begin{align*}
||\,A\,||_F = \sqrt{\sum_{i=1}^m\sum_{j=1}^n|a_{ij}|^2},
\end{align*}
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 \PMlinkescapetext{equivalent}) definition, in the case that $R=\mathbb{R}$ or $\mathbb{C}$, is
\begin{align*}
||\,A\,||_F = \sqrt{\textrm{trace}(A^*A)},
\end{align*}
where $A^*$ denotes the conjugate transpose of $A$.
Some \PMlinkescapetext{properties}:
\begin{itemize}
\item
Denote the columns of $A$ by $A_i$. A nice property of the norm is that
\begin{align*}
||A||_F^2=||A_1||_2^2+||A_2||_2^2+\cdots+||A_n||_2^2.
\end{align*}
\item Let $A$ be a square matrix and let $U$ be a unitary matrix
of same size as $A$. Then $||\,A\,||_F = ||\,U^\ast A U\,||_F$
where $U^\ast$ is the conjugate transpose of $U$.
\item If $AB$ is defined, then $||\,A B\,||_F \le ||\,A\,||_F\ ||\,B\,||_F$.
\end{itemize}