On the Moore–Penrose inverse of a sum of matrices

Abstract

The paper considers various problems concerned with the Moore–Penrose inverse of a sum of two matrices. By establishing several original results and by combining various facts known in the literature, the article reveals a number of emerging features of the inverse. The investigations shed also a spotlight on different proving approaches useful to cope with the problems in question. Rather than focusing exclusively on the main topic, the considerations endeavour to place it in a wider context, linking it with different matrix notions, facts, and tools, as well as indicating areas of its applications originating from, e.g. computational methods, physics, statistics, discriminant analysis, or neuroscience.

Keywords:

• Generalized matrix inverse
• matrix methods for engineers and scientists
• orthogonal projector
• partitioned matrix
• matrix partial ordering
• rank one modification
• matrix decomposition
• Hartwig–Spindelböck decomposition
• Schrödinger equation
• parallel sum
• electric circuits
• electroencephalography

MSC:

• 15A09
• 15A24
• 15B57

Disclosure statement

No potential conflict of interest was reported by the author(s).