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.
| Original language | English |
|---|---|
| Pages (from-to) | 143-148 |
| Number of pages | 6 |
| Journal | Linear Algebra and Its Applications |
| Volume | 538 |
| DOIs | |
| Publication status | Published - Feb 1 2018 |
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Keywords
- Generalized inverse
- Jensen convexity
- Loewner ordering
- Midpoint convexity
ASJC Scopus subject areas
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics
Cite this
A note on the convexity of the Moore–Penrose inverse. / Nordstrom Olof, Kenneth.
In: Linear Algebra and Its Applications, Vol. 538, 01.02.2018, p. 143-148.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - A note on the convexity of the Moore–Penrose inverse
AU - Nordstrom Olof, Kenneth
PY - 2018/2/1
Y1 - 2018/2/1
N2 - 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.
AB - 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.
KW - Generalized inverse
KW - Jensen convexity
KW - Loewner ordering
KW - Midpoint convexity
UR - http://www.scopus.com/inward/record.url?scp=85032037318&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85032037318&partnerID=8YFLogxK
U2 - 10.1016/j.laa.2017.10.016
DO - 10.1016/j.laa.2017.10.016
M3 - Article
VL - 538
SP - 143
EP - 148
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
SN - 0024-3795
ER -