This chapter introduces the modal matrix method for free and forced vibrations of lumped mass systems without damping. It presents several well-known eigensolution techniques of iteration method, Choleski’s decomposition, Jacobimethod, Sturm sequence method, and extraction technique. The chapter discusses the characteristics of the eigensolutions for symmetric and nonsymmetric matrices and for repeating roots. A structure must be conceived of as a model consisting of a finite number of masses connected by massless springs. The spring-mass model, depending on the characteristics of the structure, can be established in different ways. The chapter provides several general numerical methods suitable for matrix formulation. First is the iteration method, which can be applied to both symmetric and unsymmetric matrices of mass and stiffness. In the iteration method, the stiffness matrix must be inverted to a flexibility matrix, in order to obtain the fundamental mode. The inversion is time-consuming even with computer application. Matrix inversion can be avoided by using Choleski’s decomposition method.