Lecture 4: Designing linear systems; special structure (part III)

This segment details the step-by-step matrix multiplication process within the Cholesky factorization, focusing on specific elements and their relationships. The speaker highlights the importance of isolating key elements (Lkk) and explains how these relationships are crucial for the subsequent factorization algorithm. The explanation includes careful consideration of row and column vectors, addressing potential confusion regarding matrix dimensions and notation. This segment presents the core algorithm for Cholesky factorization using an inductive approach, moving row by row. The speaker clearly explains how to compute the elements of the lower triangular matrix (L) using previously computed values and forward substitution to solve a lower triangular system of equations. The explanation emphasizes the iterative nature of the algorithm, showing how each row's computation relies on the previously computed rows, making the process efficient and systematic. The lecture derives the Cholesky factorization (C = LLᵀ) for a symmetric positive definite matrix C. It uses inductive row-wise computation, solving lower triangular systems via forward substitution to find L's elements. This factorization reveals that xᵀCx represents a dot product after a linear transformation by L, highlighting the inherent Euclidean structure of positive definite matrices.