The article discusses the principle of minimized iterations in solving matrix eigenvalue problems, as interpreted by Dr. Cornelius Lanczos. The method is applied to both homogeneous and non-homogeneous characteristic matrix equations. It involves generating a series of orthogonal functions to reduce the order of the matrix equation, which can be solved directly in terms of polynomial functions. The reduced matrix equation is solved iteratively, and the convergence of the solution is observed as the order of the reduced matrix increases. The method is recommended for determining a small number of large eigenvalues and modal columns of a large matrix, offering a rapid and efficient alternative to conventional iterative procedures. The article also outlines the Lanczos algorithm and its application to both the Lanczos and Galerkin methods, highlighting the computational advantages of the minimized iterations approach.The article discusses the principle of minimized iterations in solving matrix eigenvalue problems, as interpreted by Dr. Cornelius Lanczos. The method is applied to both homogeneous and non-homogeneous characteristic matrix equations. It involves generating a series of orthogonal functions to reduce the order of the matrix equation, which can be solved directly in terms of polynomial functions. The reduced matrix equation is solved iteratively, and the convergence of the solution is observed as the order of the reduced matrix increases. The method is recommended for determining a small number of large eigenvalues and modal columns of a large matrix, offering a rapid and efficient alternative to conventional iterative procedures. The article also outlines the Lanczos algorithm and its application to both the Lanczos and Galerkin methods, highlighting the computational advantages of the minimized iterations approach.