A note on the convexity of the Moore–Penrose inverse
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Date
2018-02-01
Authors
Nordström, Kenneth
Journal Title
Journal ISSN
Volume Title
Publisher
Linear Algebra and its Applications
Abstract
Abstract This note is a sequel to an earlier study (Nordström [7]) on convexity properties of the inverse and Moore–Penrose inverse, in which the following question was raised. Given nonnegative definite matrices A and B with Moore–Penrose inverses A+ and B+, respectively, can one show that(λA+λ‾B)+⩽λA++λ‾B+ holding for a single λ∈]0,1[ is enough to guarantee its validity for all λ∈]0,1[? (The ordering above is the partial ordering, induced by the convex cone of nonnegative definite matrices, and λ‾:=1−λ.) In this note an affirmative answer is provided to this question.
Description
Keywords
Generalized inverse, Jensen convexity, Loewner ordering, Midpoint convexity
Citation
Kenneth Nordström, A note on the convexity of the Moore–Penrose inverse, In Linear Algebra and its Applications, Volume 538, 2018, Pages 143-148