# Weak group inverse

## TL;DR

This paper introduces the weak group inverse for square matrices, explores its properties, and uses it to characterize certain matrix orders, including the core-EP order.

## Contribution

The paper presents the novel weak group inverse for matrices of arbitrary index and characterizes related matrix orders using this inverse.

## Key findings

- Characterization of the weak group inverse
- Introduction of two matrix orders: pre-order and partial order
- A new characterization of the core-EP order

## Abstract

In this paper, we introduce a weak group inverse (called the WG inverse in the present paper) for square matrices of an arbitrary index, and give some of its characterizations and properties. Furthermore, we introduce two orders: one is a pre-order and the other is a partial order, and derive several characterizations of the two orders. At last, one characterization of the core-EP order is derived by using the WG inverses.