For $\mathbf{A}\in\mathbb{F}^{n\times m}$, suppose its singular values are $\sigma_1,\ldots,\sigma_{\min(n,m)}$.

Then, the Schatten $p$-norm of $\mathbf{A}$, where $p\in[1,\infty]$, is defined as [Ber09, Proposition 9.2.3]:

$\|\mathbf{A}\|_{\sigma p} \triangleq \begin{cases} \left[\sum_{i=1}^{\min(n,m)}\sigma_i^p(\mathbf{A}) \right]^{1/p}, & 1\leq p\lt\infty,\\ \sigma_\max(\mathbf{A}), & p = \infty.\end{cases}$

When $p=1$, we have the Schatten 1-norm, which is also called the trace norm or nuclear norm [Ber09, p. 549]:

$\text{tr}\langle\mathbf{A}\rangle \triangleq \|\mathbf{A}\|_{\sigma1} = \sigma_1(\mathbf{A})+\cdots+\sigma_{\min(n,m)}(\mathbf{A}).$

When $p=2$, we have the Frobenius norm:

$\|\mathbf{A}\|_F \triangleq \left[\text{tr}(\mathbf{A}^\ast\mathbf{A})\right]^{1/2} = \|\mathbf{A}\|_{\sigma2}.$

When $p=\infty$, we have the spectral norm:

$\|\mathbf{A}\|_{\sigma\infty} = \sigma_\max(\mathbf{A}).$