I am tasked to show that an antisymmetric matrix $A$ can be expressed in block diagonal form with each block being a 2x2 antisymmetric matrix or a zero submatrix. (I get that odd-dimensions can’t be expressed this way.)
The hints given in the problem were as such: 1) to consider the eigenvectors of $A^2$ (which I know is symmetric), and 2) to notice that the non-vanishing eigenvalues of $A^2$ come in pairs.
I am able to prove, via the scalar product, that two eigenvectors of $A^2$ with the same (non-zero) eigenvalues are orthogonal. However, beyond this, I am unable to move beyond this. I am unsure of how to prove hint 2), and hint 1) leads me nowhere. Could someone guide me along the right direction? Cheers!