
# Homework M

This homework is not due, nor graded, but we are happy to chat with you about the problems and discuss them on Piazza.

The idea is to have things representative of midterm problems.

There will be updates! And these have not been proofread for length or entire correctness yet.

## Problem 1

Let $\mA = \mQ \mR$.

Show that the condition number of $\mA$ is the same as the condition number of $\mR$.

(The definition of the condition number will be given on the exam.)

## Problem 2

An arrow matrix is a square matrix of the form $\mA = \mD + \vf \ve_n^T + \ve_n \vg^T$ for arbitrary $\vf$ and $\vg$ and diagonal $\mD$.

1. Give conditions such that $\mA$ is non-singular.

2. Give an algorithm to solve $\mA \vx = \vb$.

3. What is the runtime of your algorithm?

## Problem 3

Describe in words what the matrix: $\mA = \mI - \frac{1}{n} \ve \ve^T$ computes when applied to a vector $\vx$.

## Problem 4

Determine the missing element of this factorization.

## Problem 5

Is the following matrix positive definite?

## Problem 6

1. What does the following computer code compute?

a = 0;
for i=1:n
a = a + x(i)^2;
end
b = sqrt(a)

2. Is this algorithm backwards stable? (Note, I won't have any serious question chains on the exam where answer b depends on answer a.)

## Problem 7

Prove that all induced norms of a diagonal matrix are equal. (Hint, not sure if this is true, it is true for the 1-norm, 2-norm and infinity norms)