Matrix decomposition structural equation modeling (MDSEM) is introduced as a novel approach in structural equation modeling, contrasting with traditional structural equation modeling (SEM). MDSEM approximates the data matrix using a model generated by the hypothetical model and addresses limitations faced by conventional SEM procedures by emphasizing factor analysis with L[subscript 2] penalization. Key advantages of MDSEM include preventing improper solutions, the ability to compute observation-wise residuals without post-hoc factor score estimation and ease in identifying equivalent models. These benefits are attributed to its matrix decomposition techniques, allowing for direct model fitting to the data matrix, unlike the covariance structure fitting in CS-SEM. An iterative algorithm for parameter estimation is proposed, guaranteeing a monotonically decreasing function value. Theoretical properties of MDSEM are examined, revealing its shared characteristics with existing factor analysis and SEM. Numerical simulations and real data examples validate that MDSEM produces results comparable to existing methods when adequately calibrated.