Is $A$ satisfying ${A^2} = - I$ similar to $\left[ {\begin{smallmatrix} 0&I \\ { - I}&0 \end{smallmatrix}} \right]$?

Question

Is $A$ satisfying ${A^2} = - I$ similar to $\left[ {\begin{array}{*{20}{c}} 0&I \\ { - I}&0 \end{array}} \right]$?

I wonder if there is an easy way to prove this. It seems related to symplectic matrices.

Any help will be appreciated:)

Answer 1

If $A$ is defined over the reals, and $A^2=-I_n$ then $n$ is even and $A$ is similar to $\pmatrix{0&I_m\\-I_m&0}$ over the reals where $m=n/2$.

One can make $\Bbb R^{n}$ into a complex vector space via $(r+si)v =rv+sAv$. So this has dimension $m=n/2$ over $\Bbb C$. Take a basis $v_1,\ldots,v_m$ of $\Bbb R^n$ as a $\Bbb C$-vector space. Then $v_1,\ldots,v_m,Av_1,\ldots,Av_m$ is a real basis for $\Bbb R^n$. Consider the matrix of $A$ with respect to this new basis...