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

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

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.