Perturbation of Dual Group Generalized Inverse and Group Inverse

Symmetry plays a crucial role in the study of dual matrices and dual matrix group inverses. This paper is mainly divided into two parts. We present the definition of the spectral norm of a dual real matrix $\widehat{A}$, (which is usually represented in the form $\widehat{A}=A+\epsilon A_0$, A and $A_0$ are, respectively, the standard part and the infinitesimal part of $\widehat{A}$) and two matrix decompositions over dual rings. The group inverse has been extensively investigated and widely applied in the solution of singular linear systems and computations of various aspects of Markov chains. The forms of the dual group generalized inverse (DGGI for short) are given by using two matrix decompositions. The relationships among the range, the null space, and the DGGI of dual real matrices are also discussed under symmetric conditions. We use the above-mentioned facts to provide the symmetric expression of the perturbed dual real matrix and apply the dual spectral norm to discuss the perturbation of the DGGI. In the real field, we present the symmetric expression of the group inverse after the matrix perturbation under the rank condition. We also estimate the error between the group inverse and the DGGI with respect to the P-norm. Especially, we find that the error is the infinitesimal quantity of the square of a real number, which is small enough and not equal to 0.