Difference between revisions of "Frobenius matrix norm"
From Encyclopedia of Mathematics
(Importing text file) |
(TeX done) |
||
| Line 1: | Line 1: | ||
''Frobenius norm'' | ''Frobenius norm'' | ||
| − | Let | + | 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==== | |
| + | <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) {{ZBL|0413.15002}}</TD></TR> | ||
| + | </table> | ||
| − | + | {{TEX|done}} | |
| − | |||
| − | |||
| − | |||
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=11565
Frobenius matrix norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_matrix_norm&oldid=11565
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article