find complex matrices

Question

Find complex matrices satisfying the following equation

What I have is solution based on real numbers. How can I come to a solution in complex numbers?

Answer 1

If $X^{2} - \operatorname{Id} = 0$,; it means that the minimal polynomial of $X$, denoted with $m_{X}(t)$ divides the polynomial $t^{2}-1 = (t+1)(t-1)$.

Now you just have three cases to deal with, which run out the values founded with WolframAlpha, which are complex indeed, since $\mathbb{R} \subset \mathbb{C}$.

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Edit: To be more specific, you have all the conjugacy classes of the three case mentioned above. Hence you have the matrix $I$, $-I$ and all the matrix of the form $$ M \left[ \begin{matrix} 1 & 0\\ 0 & -1 \end{matrix} \right] M^{-1} $$ with $M\in \text{GL}(2,\mathbb C)$.