### 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 |
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Pages (from-to) | 143-148 |

Number of pages | 6 |

Journal | Linear Algebra and Its Applications |

Volume | 538 |

DOIs | |

Publication status | Published - Feb 1 2018 |

### Fingerprint

### 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

*Linear Algebra and Its Applications*,

*538*, 143-148. https://doi.org/10.1016/j.laa.2017.10.016

**A note on the convexity of the Moore–Penrose inverse.** / Nordstrom Olof, Kenneth.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 538, pp. 143-148. https://doi.org/10.1016/j.laa.2017.10.016

}

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 -