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characteristic map; dominant map; linear subspace; $\mathcal G\mathcal L_n$-invariant set of matrices; rank variety
We collect certain useful lemmas concerning the characteristic map, ${\mathcal GL}_n$-invariant sets of matrices, and the relative codimension. We provide a characterization of rank varieties in terms of the characteristic map as well as some necessary and some sufficient conditions for linear subspaces to allow the dominant restriction of the characteristic map.
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