Publication details

Comparative index and Lidskii angles for symplectic matrices

Authors

ŠEPITKA Peter ŠIMON HILSCHER Roman

Year of publication 2021
Type Article in Periodical
Magazine / Source Linear Algebra and its Applications
MU Faculty or unit

Faculty of Science

Citation
Web https://www.sciencedirect.com/science/article/pii/S0024379521001695#!
Doi http://dx.doi.org/10.1016/j.laa.2021.04.012
Keywords Comparative index; Dual comparative index; Lidskii angle; Argument of symplectic matrix; Conjoined basis; Lagrangian subspace
Description In this paper we establish a connection between two important concepts from the matrix analysis, which have fundamental applications in the oscillation theory of differential equations. These are the traditional Lidskii angles for symplectic matrices and the recently introduced comparative index for a pair of Lagrangian planes. We show that the comparative index can be calculated by a specific argument function of symplectic orthogonal matrices, which are constructed from the Lagrangian planes. The proof is based on a topological property of the symplectic group and on the Sturmian separation theorem for completely controllable linear Hamiltonian systems. We apply the main result in order to present elegant proofs of certain important properties of the comparative index.
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