## Subsection3.3.7Orthogonality of resulting $Q$

###### Homework3.3.7.1.

Previous programming assignments have the following routines for computing the QR factorization of a given matrix $A \text{:}$

Use these to examine the orthogonality of the computed $Q$ by writing a Matlab script (from scratch), in file Assignments/Week03/matlab/test_orthogonality.m, for the matrix

\begin{equation*} \left( \begin{array}{c | c | c } 1 \amp 1 \amp 1 \\ \epsilon \amp 0 \amp 0 \\ 0 \amp \epsilon \amp 0 \\ 0 \amp 0 \amp \epsilon \end{array} \right) . \end{equation*}
Solution ###### Ponder This3.3.7.2.

In the last homework, we examined the orthogonality of the computed matrix $Q$ for a very specific kind of matrix. The problem with that matrix is that the columns are nearly linearly dependent (the smaller $\epsilon$ is).

How can you quantify how close to being linearly dependent the columns of a matrix are?

How could you create a matrix of arbitrary size in such a way that you can control how close to being linearly dependent the columns are?

###### Homework3.3.7.3. (Optional).

Program up your solution to Ponder This 3.3.7.2 and use it to compare how mutually orthonormal the columns of the computed matrices $Q$ are.