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:
- \(A = LU\) from elimination
- \(A = QR\) from orthogonalization (Gram-Schmidt)
- \(S = Q \Lambda Q^{\mathtt{T}}\) from eigenvectors of a symmetric matrix \(S\)
- \(A = X \Lambda X^{-1}\) diagonalizes \(A\) by the eigenvector matrix \(X\)
- \(A = U \Sigma V^{\mathtt{T}} =\) (orthogonal)(diagonal)(orthogonal) = Singular Value Decomposition
Related section in textbook: I.2
Instructor: Prof. Gilbert Strang
Problems for Lecture 2
From textbook Section I.2
2. Suppose \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are column vectors with components \(a_1,\ldots,a_m\) and \(b_1,\ldots,b_p\). Can you multiply \(\boldsymbol{a}\) times \(\boldsymbol{b}^{\mathtt{T}} \) (yes or no)? What is the shape of the answer \(\boldsymbol{ab}^{\mathtt{T}} \)? What number is in row \(i\), column \(j\) of \(\boldsymbol{ab}^{\mathtt{T}} \)? What can you say about \(\boldsymbol{aa}^{\mathtt{T}} \)?
6. If \(A\) has columns \(\boldsymbol{a}_1,\boldsymbol{a}_2,\boldsymbol{a}_3\) and \(B=I\) is the identity matrix, what are the rank one matrices \(\boldsymbol{a}_1\boldsymbol{b}_1^\ast\) and \(\boldsymbol{a}_2\boldsymbol{b}_2^\ast\) and \(\boldsymbol{a}_3\boldsymbol{b}_3^\ast\) ? They should add to \(AI=A\).
