A Variational Approach to Multiparameter Eigenvalue Problems for Matrices
Paul Binding, Patrick J. Browne · SIAM Journal on Mathematical Analysis · 1977
The variational theory of eigenvalues and eigenvectors is extended to the multiparameter problem $(T_r + \sum_{s = 1}^k {\lambda _s V_{rs} } )x_r = 0$, $r = 1, \cdots ,k$ where $T_r $ and $V_{rs} $ are linear operators on finite-dimensional Hilbert spaces $H_r $. Appropriate variational problems are posed in $ \oplus _{r = 1}^k H_r $ and $ \otimes _{r = 1}^k H_r $ and give, for example, existence and reality of eigentuples and orthonormality of eigenvectors in an appropriate sense. The fact that the numerical range of an Hermitian matrix is the convex hull of its eigenvalues is directly generalized. An $\mathbb{R}^k $-valued generalized Rayleigh quotient is shown to possess analogues of constrained minimaxima and unconstrained saddle points, when evaluated at eigenvectors. Finally, dependence of $T_r $ and $V_{rs} $ on a parameter is investigated.