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If two antisymmetric and symmetric matrices are similar, why are they zero?

My work : Let $A$ be the antisymmetric matrix and $S$ the symmetric one. $A$ is diagonalizable in $M_n(\mathbb C)$, we write $\Delta_A$ its diagonal matrix. $S$ is diagonalizable in $M_n(\mathbb R)$, we write $\Delta_S$ its diagonal matrix.

$\Delta_A$ and $\Delta_S$ are similar, so they must have the same eighten values : it is impossible because some are in $\mathbb R$, the others in $\mathbb iR$. So both matrixes must be zero

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Assuming $A$ and $S$ are both real matrices, you got the right idea, but the phrasing is a bit off. "It is impossible because some are in $\mathbb R$, the others are in $\mathrm i\mathbb R$" is wrong.

What is correct is that $A$ and $S$ have the same eigenvalues as similar matrices and that those of $A$ are in $\mathrm i\mathbb R$ and those of $S$ in $\mathbb R$. Hence, the eigenvalues are all in $\mathrm i\mathbb R\cap\mathbb R = \{0\}$, so all eigenvalues are $0$.

Finally, the only diagonalizable matrix with $0$ as its only eigenvalue is the zero matrix.

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