Consider some quaternionic matrix $A$. A right eigvenvalue of $A$ is a quaternion $q$ such that $Ax=xq$ for some $x\in \mathbb{H}^n$. Similarly, a left eigenvalue of $A$ is quaternion $q$ such that $Ax=qx$ for some $x\in \mathbb{H}^n$.

I am curious if there is a generalized version of the Courant-Fischer (min-max) theorem for the right eigenvalues of Hermitian quaternionic matrices. It is known that for a Hermitian quaternionic matrix the right eigenvalues are real. This is the case I am most interested in.

For a good starting place in the references see: Fuzhen Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl., 251 (1997), 21--57. MR 1421264 (97h:15020).