AIMS Mathematics (Nov 2023)
Matrix inverses along the core parts of three matrix decompositions
Abstract
New characterizations for generalized inverses along the core parts of three matrix decompositions were investigated in this paper. Let $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $ be the core parts of the core-nilpotent decomposition, the core-EP decomposition and EP-nilpotent decomposition of $ A\in \mathbb{C}^{n\times n} $, respectively, where EP denotes the EP matrix. A number of characterizations and different representations of the Drazin inverse, the weak group inverse and the core-EP inverse were given by using the core parts $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $. One can prove that, the Drazin inverse is the inverse along $ A_{1} $, the weak group inverse is the inverse along $ \hat{A}_{1} $ and the core-EP inverse is the inverse along $ \tilde{A}_{1} $. A unified theory presented in this paper covers the Drazin inverse, the weak group inverse and the core-EP inverse based on the core parts of the core-nilpotent decomposition, the core-EP decomposition and EP-nilpotent decomposition of $ A\in \mathbb{C}^{n\times n} $, respectively. In addition, we proved that the Drazin inverse of $ A $ is the inverse of $ A $ along $ U $ and $ A_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $; the weak group inverse of $ A $ is the inverse of $ A $ along $ U $ and $ \hat{A}_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $; the core-EP inverse of $ A $ is the inverse of $ A $ along $ U $ and $ \tilde{A}_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $. Let $ X_{1} $, $ X_{4} $ and $ X_{7} $ be the generalized inverses along $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $, respectively. In the last section, some useful examples were given, which showed that the generalized inverses $ X_{1} $, $ X_{4} $ and $ X_{7} $ were different generalized inverses. For a certain singular complex matrix, the Drazin inverse coincides with the weak group inverse, which is different from the core-EP inverse. Moreover, we showed that the Drazin inverse, the weak group inverse and the core-EP inverse can be the same for a certain singular complex matrix.
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