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Suppose you have a matrix $A\in GL(2,\mathbb C)$. What are the conditions on $A$ so that $A$ is conjugated to $-A$? When $A$ is in the center, then this cannot happen. But are there some matrices such that it is true?


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$A = \left(\begin{matrix}1&0\\0&-1\end{matrix}\right)$ is conjugated to $-A$ for example... – Joel Cohen Nov 26 '12 at 21:27
up vote 1 down vote accepted

Note that $A$ and $B$ are conjugate if and only if they have the same Jordan normal form. The eigenvalues of $-A$ are exactly $-\lambda$ for all eigenvalue $\lambda$ of $A$.
Hence a necessary (but not sufficient) condition for $A$ to be conjugate to $-A$ is that for every eigenvalue $\lambda$ also $-\lambda$ is an eigenvalue of the same multiplicity. This condition is also sufficient if $A$ is diagonalizable.
If you want a few examples of such matrices, $$\begin{pmatrix}1&a\\0&-1\end{pmatrix}, \hspace{10pt}\begin{pmatrix}1&a&b\\0&0&c\\0&0&-1\end{pmatrix},\hspace{10pt}\begin{pmatrix}1&a&0&0\\0&1&0&0\\0&0&-1&0\\0&0&b&-1\end{pmatrix}$$ (For any $a,b,c$)
Further notice that if $A$ is conjugate to $-A$ then so is $P^{-1}AP$ for any invertible $P$.

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Thank you for this complete answer! – Ferenc Nov 26 '12 at 22:14

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