Consider matrices $\mathbf{A}\in\mathbb{C}^{n\times n}$ (or maybe $A\in\mathbb{R}^{n\times n}$) for which $\mathbf{A}^{-1}=-\mathbf{A}$.
A conical example (and the only one I can come up with) would be $\mathbf{A} = \boldsymbol{i}\mathbf{I},\quad \boldsymbol i^2=-1$.
Now I have a few questions about this class of matrices:
- Are there more matrices than this example matrix (I guess yes) or can they even be generally constructed somehow?
- Are there also real matrices for which this holds?
- Now each matrix that is both skew-Hermitian and unitary fulfills this property. But does it also hold in the other direction, meaning is each matrix for which $\mathbf{A}^{-1}=-\mathbf{A}$ both skew-Hermitian and unitary (maybe this is simple to prove, but I don't know where to start at the moment, but of course I know if one holds the other has to hold, too)?
- Do such matrices have any practical meaning? For example I know that Hermitian and unitary matrices are reflections (in a general sense), but what about skew-Hermitian and unitary (if 3 holds)?
This is just for personal interrest without any practical application. I just stumbled accross this property by accident and want to know more about its implications and applications.