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We show that for any rational p \in [1,\infty) except p = 1, 2, unless P =NP, there is no polynomial-time algorithm for approximating the matrix p-normto arbitrary relative precision. We also show that for any rational p\in[1,\infty) including p = 1, 2, unless P = NP, there is no polynomial-timealgorithm approximates the \infty, p mixed norm to some fixed relativeprecision.

Matrix P-norms are NP-hard to approximate if p \neq 1,2,\infty