Guide for Week 4
Math 408
-
Reading Assignment:
Homework Assignment:
-
Vocabulary Words
- Linear Least Squares Problems
- the linear least square problem
- polynomial interpolation by linear least squares
- the normal equations
- Show Null(A)=Null(A^TA)
- uniqueness in the normal equations
- orthogonal projections
- QR factorization
- orthogonal projections onto the 4 fundamental subspaces
- how to solve the normal equations using the QR factorization
- Optimization of Quadratic functions
- quadratic functions
- the relationship between linear least squares and quadratic functions
- symmetric and self-adjoint matrices
- unitary matrices
- diagonal matrices
- positive/negative definite/semi-definite matrics
- Lemma 3.1 page 29 of the course notes.
- necessary and sufficient conditions for there to exist a optimal solutions to the
quadratic optimization problems with and without affine constraints
- necessary and sufficient conditions for there to exist a unique optimal solution to the
quadratic optimizations problem with and without affine constraints
- the optimal value in a quadratic optimizations problem when an optimal solution does not exist
- Choleski factorization
- the generalized Choleski factorization
- the conjugate gradient algorithm
-
Key Concepts:
- Linear Least Squares Problems
- the normal equations
- orthogonal projections
- the QR factorization
- Optimization of Quadratic functions
- the relationship between linear least squares and quadratic functions
- positive definite/semi-definite matrices
- the existence and uniqueness theorem for quadratic optimization
- Cholesky factorization
-
Skills to Master:
- Forming and solving the normal equations
- Computing orthogonal projections
- Computing the QR factorization
- using the QR factorization to solve the normal equations
- solving a quadratic optimization problem
- computing Cholesky factorizations
-
Quiz:
-
The quiz will be on the material from Chapter 3 sections 1, 2 and 3.
The focus of the first question will center around Theorems 1.1, 2.1 and 2.2 as well as their
attendant definitions and propositions. Issues and consequences of Lemma 3.1 may also be included.
The second question will be computational in nature and will be similar to questions 1-5
on homework 3 and questions 3 and 7 on homework 4. I will not ask a question on the
Cholesky factorization or the conjugate algorithm.