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Difference between revisions of "Frobenius matrix norm"

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''Frobenius norm''
 
''Frobenius norm''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130260/f1302601.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130260/f1302602.png" />-matrix, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130260/f1302603.png" /> be the norm in the underlying field (usually <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130260/f1302604.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130260/f1302605.png" /> with the standard norm (absolute value)). The Frobenius norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130260/f1302606.png" /> is defined as
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Let $A$ be an $(m \times n)$-matrix, and let $\vert\cdot\vert$ be the norm in the underlying field (usually $\mathbf R$ or $\mathbf C$ with the standard norm ([[absolute value]])). The Frobenius norm of $A$ is defined as
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$$
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\Vert A \Vert^2_F = \sum_{i,j} \vert a_{i,j}\vert^2 \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130260/f1302607.png" /></td> </tr></table>
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Note that this norm differs from the operator norm of $A$ (for instance because $\Vert I_n \Vert_F = n$; cf. [[Norm]]). If $U$ and $V$ are unitary matrices of appropriate size,
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$$
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\Vert UAV \Vert_F = \Vert A \Vert_F \ .
 +
$$
  
Note that this norm differs from the operator norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130260/f1302608.png" /> (for instance because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130260/f1302609.png" />; cf. [[Norm|Norm]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130260/f13026010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130260/f13026011.png" /> are unitary matrices of appropriate size,
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====References====
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Noble,  J.W. Daniels,  "Applied linear algebra" , Prentice-Hall  (1969)  pp. 328ff  (Edition: Second) {{ZBL|0413.15002}}</TD></TR>
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</table>
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130260/f13026012.png" /></td> </tr></table>
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{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Noble,  J.W. Daniels,  "Applied linear algebra" , Prentice-Hall  (1969)  pp. 328ff  (Edition: Second)</TD></TR></table>
 

Revision as of 17:42, 23 May 2017

Frobenius norm

Let $A$ be an $(m \times n)$-matrix, and let $\vert\cdot\vert$ be the norm in the underlying field (usually $\mathbf R$ or $\mathbf C$ with the standard norm (absolute value)). The Frobenius norm of $A$ is defined as $$ \Vert A \Vert^2_F = \sum_{i,j} \vert a_{i,j}\vert^2 \ . $$

Note that this norm differs from the operator norm of $A$ (for instance because $\Vert I_n \Vert_F = n$; cf. Norm). If $U$ and $V$ are unitary matrices of appropriate size, $$ \Vert UAV \Vert_F = \Vert A \Vert_F \ . $$

References

[a1] B. Noble, J.W. Daniels, "Applied linear algebra" , Prentice-Hall (1969) pp. 328ff (Edition: Second) Zbl 0413.15002
How to Cite This Entry:
Frobenius matrix norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_matrix_norm&oldid=41548
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article