# Generalized Courant-Fischer theorem

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).

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I found an interesting paper: M. Seetharama Gowda and J. Tao, The Cauchy interlacing theorem in simple Euclidean Jordan algebras and some consequences. Which can be found here: math.umbc.edu/~gowda/tech-reports/trGOW08-02.pdf. However I am looking for a form of the Courant-Fischer theorem in which we would take maxs/mins over vectors in H^n. –  hypercube Dec 10 '10 at 4:47