| Version 12 |
Version 11 |
| Let $R$ be an ordered 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 |
Let $R$ be an ordered 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*} |
\begin{align*} |
| ||\,A\,||_F = \sqrt{\sum_{i=1}^m\sum_{j=1}^n|a_{ij}|^2}, |
||\,A\,||_F = \sqrt{\sum_{i=1}^m\sum_{j=1}^n|a_{ij}|^2}, |
| \end{align*} |
\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 is |
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 is |
| \begin{align*} |
\begin{align*} |
| ||\,A\,||_F = \sqrt{\textrm{trace}(A^*A)}, |
||\,A\,||_F = \sqrt{\textrm{trace}(A^*A)}, |
| \end{align*} |
\end{align*} |
| where $A^*$ denotes the conjugate transpose of $A$. |
where $A^*$ denotes the conjugate transpose of $A$. |
|
|
| Some \PMlinkescapetext{properties}: |
Some \PMlinkescapetext{properties}: |
| \begin{itemize} |
\begin{itemize} |
| \item |
\item |
| Denote the columns of $A$ by $A_i$. A nice property of the norm is that |
Denote the columns of $A$ by $A_i$. A nice property of the norm is that |
| \begin{align*} |
\begin{align*} |
| ||A||_F^2=||A_1||_2^2+||A_2||_2^2+\cdots+||A_n||_2^2. |
||A||_F^2=||A_1||_2^2+||A_2||_2^2+\cdots+||A_n||_2^2. |
| \end{align*} |
\end{align*} |
| \item Let $A$ be a square matrix and let $U$ be a unitary matrix |
\item Let $A$ be a square matrix and let $U$ be a unitary matrix |
| of same size as $A$. Then $||\,A\,||_F = ||\,U^\ast A U\,||_F$ |
of same size as $A$. Then $||\,A\,||_F = ||\,U^\ast A U\,||_F$ |
| where $U^\ast$ is the conjugate transpose of $U$. |
where $U^\ast$ is the conjugate transpose of $U$. |
| \item If $AB$ is defined, then $||\,A B\,||_F \le ||\,A\,||_F\ ||\,B\,||_F$. |
\item If $AB$ is defined, then $||\,A B\,||_F \le ||\,A\,||_F\ ||\,B\,||_F$. |
| \end{itemize} |
\end{itemize} |
