Abstract
Abstract
We investigate some geometric properties of the real algebraic variety
$$\Delta $$
Δ
of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart–Young–Mirsky-type theorem for the distance function from a generic matrix to points in
$$\Delta $$
Δ
. We exhibit connections of our study to real algebraic geometry (computing the Euclidean distance degree of
$$\Delta $$
Δ
) and random matrix theory.
Funder
Max Planck Institute for Mathematics in the Sciences
Publisher
Springer Science and Business Media LLC
Reference36 articles.
1. Agrachev, A.A.: Spaces of symmetric operators with multiple ground states. Funktsional. Anal. i Prilozhen. 45(4), 1–15 (2011)
2. Agrachev, A.A., Lerario, A.: Systems of quadratic inequalities. Proc. Lond. Math. Soc. (3) 105(3), 622–660 (2012)
3. Arnold, V.I.: Topological properties of eigen oscillations in mathematical physics. Tr. Mat. Inst. Steklova 273(Sovremennye Problemy Matematiki), 30–40 (2011)
4. Arnold, V.I.: Modes and quasimodes. Funkcional. Anal. i Priložen. 6(2), 12–20 (1972)
5. Arnold, V.I.: Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect. Sel. Math. (N.S.) 1(1), 1–19 (1995)
Cited by
15 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献