New classes of more general weighted outer inverses

Abstract

According to known definition in the literature, the W-weighted outer inverses of A is defined as the outer inverses of WAW. We introduce and study more general weighted outer inverses of rectangular complex matrices. Introduced families of outer inverses include previous research in two aspects from two approaches. In the first approach, for given complex matrices A, M, N, B, C, we define the $(M,N)$-weighted $(B,C)$-inverse of A and prove that it is just the outer inverse of MAN with the range $\mathcal{R}\left(B\right)$ and null space $\mathcal{N}\left(C\right)$. Thus, the $(M,N)$-weighted $(B,C)$-inverse generalizes the notion of the weighted outer inverse and, finally, the notion of the outer inverse with known image and kernel. We give equivalent conditions for the existence and representations of the $(M,N)$-weighted $(B,C)$-inverse. Integral and limit representations of weighted generalized inverses can be derived as corollaries. Inspired by corresponding representations of the W-weighted Drazin inverse and the weighted core-EP inverse, in the second approach, we investigate the expressions $A\left[\right(WA{)}_{\mathcal{R}\left(E\right),\mathcal{N}\left(F\right)}^{\left(2\right)}{]}^2$, $\left[\right(AW{)}_{\mathcal{R}\left(U\right),\mathcal{N}\left(V\right)}^{\left(2\right)}{]}^{2}A$ and ${\left(AW\right)}_{\mathcal{R}\left(U\right),\mathcal{N}\left(V\right)}^{\left(2\right)}A{\left(WA\right)}_{\mathcal{R}\left(E\right),\mathcal{N}\left(F\right)}^{\left(2\right)}$. Particularly, conditions under which these expressions become W-weighted outer inverses of A are analysed.

Keywords:

• Weighted outer inverse
• outer inverse
• Drazin inverse
• core-EP inverse
• representation

2010 Mathematics Subject Classifications:

• 15A09
• 65F20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This project is supported by the bilateral project between China and Serbia ‘The theory of tensors, operator matrices and applications (no. 4-5)’. Predrag Stanimirović gratefully acknowledge support from the Ministry of Education, Science and Technological Development, Republic of Serbia [grant number 174013], and support from Shanghai Key Laboratory of Contemporary Applied Mathematics. Dijana Mosić is supported by the Ministry of Education, Science and Technological Development, Republic of Serbia [grant number 174007]. Haifeng Ma is supported by the bilateral project between China and Poland (No. 37-18) and the National Natural Science Foundation of China [grant number 11971136].