Stable perturbations of matrices over rings with respect to pseudo core inverses
Liyun Wu, Jianlong Chen · Georgian Mathematical Journal · 2025
Abstract We introduce the stable perturbation of matrices over an arbitrary ∗ {\ast} -ring with respect to pseudo core inverses, which extends the stable perturbation of square complex matrices with respect to core-EP inverses. Let A and B be pseudo core invertible matrices over a unital ∗ {\ast} -ring. Then B is called a stable perturbation of A with respect to pseudo core inverses if ℛ ( A k ) ∩ 𝒩 ( ( B s ) ∗ ) = { 0 } {\mathcal{R}(A^{k})\cap\mathcal{N}((B^{s})^{\ast})=\{0\}} and ℛ ( B s ) ∩ 𝒩 ( ( A k ) ∗ ) = { 0 } {\mathcal{R}(B^{s})\cap\mathcal{N}((A^{k})^{\ast})=\{0\}} , where k and s are pseudo core indices of A and B, respectively. It is shown that if R is a right FP-injective ring, then B is a stable perturbation of A if and only if I - ( B π - A π ) 2 {I-(B^{\pi}-A^{\pi})^{2}} is invertible, where A π = I - A A \⃝raisebox{0.569055pt}{