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'Frobenius matrix norm'
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| Title of object: |
Frobenius matrix norm |
| Canonical Name: |
FrobeniusMatrixNorm |
| Type: |
Definition |
| Created on: |
2001-10-06 14:43:07 |
| Modified on: |
2004-08-24 13:12:03 |
| Classification: |
msc:15A60, msc:65F35 |
| Synonyms: |
Frobenius matrix norm=euclidean matrix norm
Frobenius matrix norm=matrix F-norm
Frobenius matrix norm=Hilbert-Schmidt norm |
Revision comment (for changes between this and next version):
| Changes for correction #5675 ('unitary transformation'). |
Preamble:
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Content:
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 $||A||_F:M(R)\ra\R$ given by
\begin{align*}
||\,A\,||_F = \sqrt{\sum_{i=1}^m\sum_{j=1}^n|a_{ij}|^2}
\end{align*}
A more concise (though equivalent) definition is
\begin{align*}
||\,A\,||_F = \sqrt{\textrm{trace}(A^*A)},
\end{align*}
where $A^*$ denotes the conjugate transpose of $A$.
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*}
(see \PMlinkname{trace}{Trace}, \PMlinkname{transpose}{Transpose}) |
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