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\|\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}