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

[a1]	B. Noble, J.W. Daniels, "Applied linear algebra" , Prentice-Hall (1969) pp. 328ff (Edition: Second) Zbl 0413.15002