Numerical Linear Algebra              Spring 2025

Homework Assignments

Homework 1: Due Friday, January 31

My problems, HW1

and

Golub & Van Loan (G&VL): P2.7.1 (p. 102). Hint: (2.7.8) refers to an algorithm on p. 98, and Lemma 2.7.1 (using interpretation (2.7.12)) does the hard part.

Homework 2: Due Friday, February 14

G&VL: P1.1.3, P1.1.4 (you can skip the last question in P1.1.4), P2.1.2

53903 students: G&VL: P3.2.4

Homework 3: Due Friday, March 14

P2.1.5, P2.2.7, P2.3.8, P2.6.1 (assume ||AB|| ≤ ||A||||B||), P2.6.2 (assume ||AB|| ≤ ||A||||B||)

Homework 4: Due Friday, April 11

GSPuzzle

G&VL: P5.1.1

Householder exercise

Homework 5: Due Wednesday, April 30

1. If A has eigenpairs (λ_j,v_j), what are the eigenpairs of the matrix A² ?

2. If A has eigenpairs (λ_j,v_j) and s is a scalar, what are the eigenpairs of the matrix A - sI? Show how you got this.

3. Show that if A is invertible, s a scalar, and Ax = sx, then 1/s is an eigenvalue of A^-1. What is its associated eigenvector?

4. Show that if t is not an eigenvalue of A, and (A-tI)^-1x = sx, then x is an eigenvector of A-tI. What is the eigenvalue of A associated with x?

5. If x is an approximate eigenvector of A, then find a scalar s that minimizes the residual ||Ax-sx||₂.