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A continuous orbit for the generalized inverse, Moore–Penrose inverse and group inverse
Linear and Multilinear Algebra ( IF 0.9 ) Pub Date : 2020-12-22 , DOI: 10.1080/03081087.2020.1861176
Saijie Chen 1 , Qianglian Huang 1 , Lanping Zhu 1
Affiliation  

As is well known, the generalized inverse, Moore–Penrose inverse and group inverse are not continuous, i.e. for θ = { 1 , 2 } , { 1 , 2 , 3 , 4 } and { 1 , 2 , 5 } , a linear bounded operator T has a θ-inverse T θ , the perturbed operator T ¯ = T + δ T is not necessary θ-invertible and even if it is θ-invertible, lim δ T 0 T ¯ θ = T θ may not be true. In this paper, we prove that T + T T θ δ T T θ T is θ-invertible and its θ-inverse ( T + T T θ δ T T θ T ) θ has the simplest possible expression, which satisfies lim δ T 0 ( T + T T θ δ T T θ T ) θ = T θ . Thus, we have found a continuous orbit for the generalized inverse, Moore–Penrose inverse and group inverse.



中文翻译:

广义逆,摩尔-彭罗斯逆和群逆的连续轨道

众所周知,广义逆,摩尔-彭罗斯逆和群逆不是连续的,即对于 θ = { 1个 2 } { 1个 2 3 4 } { 1个 2 5 } ,线性有界算子T具有θ Ť θ ,受干扰的运算符 Ť ¯ = Ť + δ Ť 不必是θ可逆的,即使它是θ可逆的, δ Ť 0 Ť ¯ θ = Ť θ 可能不是真的。在本文中,我们证明 Ť + Ť Ť θ δ Ť Ť θ Ť θ-可逆且θ- Ť + Ť Ť θ δ Ť Ť θ Ť θ 具有最简单的表达式,可以满足 δ Ť 0 Ť + Ť Ť θ δ Ť Ť θ Ť θ = Ť θ 。因此,我们发现了广义逆,摩尔-彭罗斯逆和群逆的连续轨道。

更新日期:2020-12-22
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