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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.