This lecture covers the concept of low rank approximations, focusing on the mathematical theory behind it and its applications. The instructor explains the process of finding the best approximation of a matrix by a low-rank matrix, emphasizing the importance of spectral theorems and orthogonality. Various demonstrations and examples are provided to illustrate the theory, including the minimization of certain functions. The lecture concludes with a discussion on the practical implications of low rank approximations in data analysis and signal processing.