Suppose that $A \in \mathbb{R}^{n \times n}$ is an orthogonal matrix. Then the matrix norm induced by the Euclidean norm satisfies $\| A \|_2 = 1$. Are there similar results for matrix norms $\| A \|_p$ induced by $p$-norms with $p \in \mathbb{N}$?
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1$\begingroup$ No, they are norms generally speaking $\endgroup$– Toni MhaxCommented Jul 9, 2019 at 13:50
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$\begingroup$ Why would you expect anything to hold for $p\neq 2$ ? $\endgroup$– Gabriel RomonCommented Jul 9, 2019 at 14:28
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$\begingroup$ You need some notion of orthogonality. $\endgroup$– copper.hatCommented Jul 9, 2019 at 14:51
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$\begingroup$ @copper.hat: By assumption $A$ is an orthogonal matrix. $\endgroup$– user355419Commented Jul 9, 2019 at 15:30
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$\begingroup$ @GabrielRomon: I don't have any expectations. $\endgroup$– user355419Commented Jul 9, 2019 at 15:31
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