A necessary condition for certain functions to preserve positive semi-definiteness on partitioned matrices

2016/12/12 by Lutz Klotz, Klotz, Lutz, Conrad Mädler +1
Computer Science · Mathematics · #15A15 #15B57 #Advanced Optimization Algorithms Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematics and Applications #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.1612.03660

openalex publication_date 2016/12/12 · openalex created_date 2017/01/06 · openalex updated_date 2026/07/28

Abstract

If f is a symmetric complex-valued function on the m-fold Cartesian product of the set of non-negative reals and A is a positive semi-definite m× m matrix with eigenvalues λ_j, we set f(A):=f(λ₁,\dotsc,λ_m). It is shown that if [f(A_αβ)] is positive semi-definite whenever [A_αβ] is a positive semi-definite matrix with positive semi-definite entries A_αβ, then f has a power series expansion with positive coefficients.

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