Supports for minimal hermitian matrices

2019/06/14 by Mendoza, Alberto, Recht, Lázaro, Varela, Alejandro
#14M15 #15A12 #15B57 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1906.06417

Abstract

We study certain pairs of subspaces V and W of ℂⁿ we call supports that consist of eigenspaces of the eigenvalues ±‖M‖ of a minimal hermitian matrix M (‖M‖≤ ‖M+D‖ for all real diagonals D). For any pair of orthogonal subspaces we define a non negative invariant δ called the adequacy to measure how close they are to form a support and to detect one. This function δ is the minimum of another map F defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of F in order to approximate δ. These results allow us to prove that the set of supports has interior points in the space of flag manifolds.

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