A note on the convexity of the Moore-Penrose inverse

作者：Nordstrom Kenneth^*

来源：Linear Algebra and Its Applications, 2018, 538: 143-148.

DOI：10.1016/j.laa.2017.10.016

• Generalized inverse
• Jensen convexity
• Loewner ordering
• Midpoint convexity

This note is a sequel to an earlier study (Nordstrom [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
(lambda A + (lambda) over barB)(+) <= lambda A(+) (lambda) over barB(+)
holding for a single lambda is an element of 10,1] is enough to guarantee its validity for all lambda E]0, 1[? (The ordering above is the partial ordering, induced by the convex cone of nonnegative definite matrices, and (lambda) over bar : = 1 - lambda.) In this note an affirmative answer is provided to this question.

• 出版日期2018-2-1