Frobenius norm

Remarks

Equivalently, $\norm{A}_F = \sqrt{\tr(A^* A)}$, where $A^*$ is the conjugate transpose and $\tr$ is the trace. If $\sigma_1, \ldots, \sigma_r$ are the nonzero singular values of $A$, then $\norm{A}_F^2 = \sigma_1^2 + \cdots + \sigma_r^2$. Example: for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, $\norm{A}_F = \sqrt{30}$. The Frobenius norm is a submultiplicative matrix norm, by the Cauchy–Schwarz inequality, and $\norm{UAV}_F = \norm{A}_F$ for unitary $U$ and $V$. It is not an operator norm when $n \ge 2$: every operator norm gives $\norm{I_n} = 1$, but $\norm{I_n}_F = \sqrt{n}$.