Lecture 2: Multiplying and Factoring Matrices

Description

Multiplying and factoring matrices are the topics of this lecture. Professor Strang reviews multiplying columns by rows: \(AB =\) sum of rank one matrices. He also introduces the five most important factorizations.

Summary

Multiply columns by rows: \(AB =\) sum of rank one matrices

Five great factorizations:

1. \(A = LU\) from elimination
2. \(A = QR\) from orthogonalization (Gram-Schmidt)
3. \(S = Q \Lambda Q^{\mathtt{T}}\) from eigenvectors of a symmetric matrix \(S\)
4. \(A = X \Lambda X^{-1}\) diagonalizes \(A\) by the eigenvector matrix \(X\)
5. \(A = U \Sigma V^{\mathtt{T}} =\) (orthogonal)(diagonal)(orthogonal) = Singular Value Decomposition

Related section in textbook: I.2

Instructor: Prof. Gilbert Strang