aspg-33-2057-1792223085 20261017074445 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Galoitica: Journal of Mathematical Structures and Applications GJMSA 2834-5568 2023 7 2 Lee Xu University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China This paper is dedicated to find Legendre polynomials by using novel linear algebraic methods based on matrices based on Liouville-Sturm theorem, where we find the matrix of the differential operator for Legendre polynomials, with their eigenvalues and their eigenvectors. Also, we illustrate many examples to clarify the validity of our work. 2023 47 50 2057 https://creativecommons.org/licenses/by/4.0/ 10.54216/GJMSA.070205 https://americaspg.com/journal/33/article/2057 https://americaspg.com/storage/11694238573.pdf