Extensions for Generalized Inverse of Matrices

The theory of generalized inverse of matrices has been extensively studied due to its powerful capability of solving linear equations system and singular value problems.In this paper,we extend the generalized inverse of matrices to the setting of Euclidean Jordan algebra.We first introduce and characterize the concept of generalized inverse of the element in Euclidean Jordan algebra.We then introduce and characterize the concept of generalized inverse of Lyapunov transformation in Euclidean Jordan algebra.Finally,we indicate some potential applications for the theory of generalized inverse in Euclidean Jordan algebra.