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In this short article, for any matrix $X\in\mathbb{R}^{n\times m}$ theproximity operator of two induced norms $ \|X\|_1 $ and $ \|X\|_{\infty}$ arederived. Although no close form expression is obtained, an algorithmicprocedure is described which costs roughly $\mathcal{O}(nm)$. This algorithmrelies on a bisection on a real parameter derived from the Karush-Kuhn-Tuckerconditions, following the proof idea of the proximal operator of the $ \max $function found in Parikh(2014).

Computing the proximal operator of the $\ell_1$ induced matrix norm