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A function $f$ on $M_{N\times n}\left(ℝ\right)$ which is $\mathrm{SO}\left(N\right)\times \mathrm{SO}\left(n\right)$-invariant is convex if and only if its restriction to the subspace of diagonal matrices is convex. This results from Von Neumann type inequalities and appeals, in the case where $N=n$, to the notion of .