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In this note, we consider matrices similar to $X$-form matrices, which arethe matrices for which only the diagonal and the anti-diagonal elements can bedifferent from zero. First, we give a characterization of these matrices usingthe minimal polynomial. Then, we prove that the set of matrices similar to$X$-form matrices over $\mathbb{R}$ and $\mathbb{C}$ are dense and we give acharacterization of the interior of this set.

A dense subset of $M\_{n}(\mathbb{R})$ containing diagonalizable matrices