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The $p$-norm of a matrix $A$ for $p \geq 1$ is defined as $$\|A\|_p = \max_{ x \in \mathbb{R}^n, \|x\|_p=1} \|Ax\|_p.$$ My question: does this equal $$ \max_{ x \in \mathbb{C}^n, \|x\|_p=1} \|Ax\|_p$$ The only difference is the replacement of $\mathbb R^n$ by $\mathbb C^n$. The answer is certainly yes for $p=1,2,\infty$ and the Wikipedia article on matrix norms appears to imply the answer is yes for any $p$.

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