Study guide for Midterm 2

Here is a non-exhaustive list of questions you should be able to answer as you prepare for the first midterm. The midterm will cover chapter 1-4.

Linear Least Squares

How can you solve a least-squares problem using the SVD?
Given an SVD of the matrix and a right-hand side, how would you find the 2-norm of the residual of a least-squares problem?
How can least squares problems be solved via the normal equations? What are the advantages and disadvantages of that?
On what factors does the conditioning of a linear least-squares problem depend on?
What is QR factorization, when does it exist and is it unique?
What is an orthogonal matrix?
What is a projection matrix?
What are the classical and modified Gram-Schmidt processes? What can you say about their stability?
What is a Householder reflector matrix, what properties does it have and how is it computed?
What is the Householder QR factorization algorithm, and what can you say about its stability?
How many Householder iterations are needed for a QR factorization and what is the total cost?
What is a Givens rotation matrix, what properties does it have and how is it computed?
How can Givens rotations be used to factorize a sparse matrix?
How does one solve linear least squares problems using a QR factorization?
What is Cholesky QR and how is it used for solving a linear least squares problem?
How can you solve a rank-deficient linear least squares problem?

What is an eigenvector? an eigenvalue of a matrix? (i.e. know the definition)
What is a similarity transformation?
What is the relationship between the SVD and the eigenvalue decomposition?
When are eigenvectors linearly independent?
What are the Jordan and Schur forms?
What is a normal matrix? a defective matrix? a diagonalizable matrix?
What is an eigenvalue multiplicity? what is a complex eigenvalue pair?
What is the relationship between the SVD and the eigenvalue decomposition?
What is power iteration? What can be obtained using power iteration?
What is normalized power iteration? What problem does it address?
Given an approximate eigenvector, how can you estimate eigenvalues?
What is the Rayleigh Quotient? Rayleigh Quotient iteration?
What is inverse iteration and inverse shift iteration? Which eigenvalue does it find?
What is the conditioning of eigenvalues and eigenvectors in an eigenvalue problem?
What are the eigenvalues of an upper triangular matrix and how do you compute the eigenvectors?
What is orthogonal iteration? QR iteration? how are they related?
How can one reduce a matrix to Hessenberg form and why is it helpful?
How can one incorporate shifting into QR iteration?
What is deflation?
What is a Krylov subspace? how is it related to a Companion matrix?
What is the Arnoldi method? the Lanczos method?