中文摘要：

广义逆矩阵在处理线性方程组与奇异值问题中的强大能力，使得这一理论得到广泛应用．本文将矩阵的广义逆推广到欧几里德若当代数中．首先，引入并刻画了欧几里德若当代数中元素的广义逆．然后，对该代数中一类重要的线性变换：Lyapunov变换的广义逆进行了刻画．最后，指出了欧几里德若当代数中广义逆理论的某些潜在应用．

英文摘要：

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.