The -positive semidefinite matrices A such that are nonsingular

Abstract

This work focuses on the class of $\mathcal{L}$-positive semidefinite matrices A such that $A{A}_{(\mathcal{L})}^{(†)}-A_{(\mathcal{L})}^{(†)}A$ is nonsingular, where $A_{(\mathcal{L})}^{(†)}$ is the generalized Bott-Duffin inverse of A with respect to a subspace $\mathcal{L}$. We give some equivalent characterizations for the nonsingularity of $A{A}_{(\mathcal{L})}^{(†)}-A_{(\mathcal{L})}^{(†)}A$, $A{A}_{(\mathcal{L})}^{(†)}+A_{(\mathcal{L})}^{(†)}A$ and $I_n-A(A_{(\mathcal{L})}^{(†)}{)}^{2}A$ mainly by rank equalities and subspace operations, among which relations are pointed out. Meanwhile we show the inverses of their general versions in a decomposition form. In addition, the continuity of this class of matrices is discussed based on the continuity of the generalized Bott–Duffin inverse.

Keywords:

• Generalized Bott–Duffin inverse
• $\mathcal{L}$-positive semidefinite matrices
• $\mathrm{co}$-$\mathrm{GBD}$ matrices
• nonsingularity
• continuity

AMS Classifications:

• 15A03
• 15A09
• 15B57

Acknowledgments

The authors are very grateful to our handing editor and the anonymous referee for valuable suggestions and comments, which improved the presentation of the paper distinctly.

Disclosure statement

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

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 11961076].