Suppose I have a diagonal matrix such as
$$ A:=
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1 \\
\end{bmatrix},
$$
Is there a way to find an invertible matrix P, such that: $-A = P^{-1} A P $?
2025-04-15 09:03:48.1744707828
Can this diagonal matrix be similar to it's negation?
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1
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No.
The existence of such a matrix $P$ would imply that $-A$ is similar to $A$. In particular, this would imply that $$ \DeclareMathOperator{trace}{trace}\trace(A)=\trace(-A) $$ which is of course impossible since $\trace(A)=1$ and $\trace(-A)=-1$.