Spectral (Isotropic) Manifolds and Their Dimension

Aris Danilidis (1, 2) , Jérôme Malick (3) , Hristo Sendov (4)

1 UAB - Departament de Matemàtiques [Barcelona]
2 DIM - Departamento de Ingeniería Matemática [Santiago]
3 DAO - Données, Apprentissage et Optimisation
4 Department of Statistics and Actuarial Science [Toronto]

A set of symmetric matrices whose ordered vector of eigenvalues belongs to a fixed set in Rn is called spectral or isotropic. In this paper, we establish that every locally symmetric submanifold M of Rn gives rise to a spectral manifold, for k ∈ {2, 3, . . . , ∞, ω}. An explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of M is derived.

Mots clés: locally symmetric set; eigenvalue; symmetric matrix; invariant function; spectral manifold; isotropic set; invariant function.

Domaines: Optimisation et contrôle [math.OC]; Géométrie différentielle [math.DG]

Aris Danilidis, Jérôme Malick, Hristo Sendov. Spectral (Isotropic) Manifolds and Their Dimension. Journal d'analyse mathématique, 2016, 128 (1), pp.369-397. ⟨10.1007/s11854-016-0013-0⟩. ⟨hal-00972281⟩