What's the condition for a matrix $A$ ($2N\times 2N$ dimension) to have eigenvalues in pairs $\pm\lambda$? 
For a given even dimension square complex matrix $A$ ($2N\times 2N$ dimension),
  what's the sufficient and necessary condition for the matrix $A$ such that:
  if $\lambda_{1}$ is an eigenvalue, then $\lambda_{2}=-\lambda_{1}$ is also an eigenvalue? 

I have found one sufficient condition: skew-symmetric matrix, but
obviously there is a lot of matrices having eigenvalues in pairs are
not skew-symmetric. 
So I wonder is there a sufficient and necessary condition? 
Or is there a name for this family of matrices (skew-symmetric matrix
is a member)? 

Take any Hermitian matrix of the form $\mathfrak{B}=\left[\begin{array}{c|c}\mathbf{0}_n & A\\\hline A^* & \mathbf{0}_m\end{array}\right],$ where $A$ is $n\times m$. Then if $\lambda$ is an eigenvalue of $\mathfrak{B}$ with multiplicity $k$, then $-\lambda$ is also an eigenvalue with the same multiplicity. This is another sufficient condition. 
