Perturbation estimation for the parallel sum of Hermitian positive semi-definite matrices

ABSTRACT

Let ${\mathbb{C}}^{n\times n}$ be the set of all $n\times n$ complex matrices. For any Hermitian positive semi-definite matrices A and B in ${\mathbb{C}}^{n\times n}$, their new common upper bound less than $A+B-A:B$ is constructed, where ${(A+B)}^{†}$ denotes the Moore–Penrose inverse of $A+B$, and $A:B=A{(A+B)}^{†}B$ is the parallel sum of A and B. A factorization formula for $(A+X):(B+Y)-A:B-X:Y$ is derived, where $X,Y\in {\mathbb{C}}^{n\times n}$ are any Hermitian positive semi-definite perturbations of A and B, respectively. Based on the derived factorization formula and the constructed common upper bound of X and Y, some new and sharp norm upper bounds of $(A+X):(B+Y)-A:B$ are provided. Numerical examples are also provided to illustrate the sharpness of the obtained norm upper bounds.

KEYWORDS:

• Moore–Penrose inverse
• parallel sum
• perturbation estimation
• norm upper bound

AMS SUBJECT CLASSIFICATIONS:

• 15A09
• 15A60
• 46L05
• 47A55

Acknowledgements

The authors thank the referee for helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11671261].