I analyse real-world data by decomposing asymmetric square matrices with zero diagonals into a symmetric and a skew-symmetric part, and treating the eigenstructure of the skew-symmetric part as providing canonical variates, in a method proposed by Gower. I find this provides often strikingly interpretable results and am working towards encapsulating it with additional analyses in a piece of software. For instance, simple rotation of the axes often produces instantly interpretable patterns. The first two eigenvalues typically, but not always, account for 98% of the variance in the matrix.
Does the fact that the eigenvalues are purely imaginary have any practical consequences that may not be apparent to a non-mathematician long used to working with real positive definite matrices?
The differential equation $$ \dot x(t) + Ax(t)=0 $$ is norm-conserving (energy-conserving) for skew-symmetric $A$ in the sense that $$ \|x(t)\|_2= \|x(0)\|_2 $$ for all $t$. This might be compared to $|\exp(it)|=1$ for all $t$.
If $A$ would be positive definite the $\|x(t)\|_2$ would decay exponentially fast.