View source for Frobenius matrix norm

''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====
<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}}