TITLE: If $A$ is similar to $-A^T$, is $A$ similar to an antisymmetric matrix?
QUESTION [5 upvotes]: Suppose that $A$ is an $n \times n$-matrix with entries from some field $F$. Assume also that $A$ is similar to $-A^T$, ie. $PAP^{-1} = -A^T$ for some invertible matrix $P$. (Here $A^T$ denotes the transpose of $A$.) Does it follow that $A$ is similar to an antisymmetric matrix?

REPLY [6 votes]: Consider the matrix $A = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.  Since $A$ is diagonal, it is similar to $A^T$ (trivially), and since $A$ and $-A$ have the same eigenvalues, they are similar.  Thus $A$ is similar to $-A^T$.
However the eigenvalues of a real antisymmetric matrix are purely imaginary, so $A$ is not similar to one of these.