Find Condition Number

This example goes over finding the condition number given a series of matrices.

Problem 1:

Find the condition number of given the following:
stretch_min_max.png

Solution:

Here we can see that the min-stretch is and the max-stretch is so .

Problem 2:

Find the condition number of the matrix
Where is any angle in radians.

Soultion:

Since the rotation matrix would be rotating the unit circle counterclockwise by the condition number is

Problem 3:

What is ?

Solution:

Start with such that , what happens to ?

To start we realize that is a rotation by so:

example_cond_num_A4.png

Next we realize that is stretching by in the direction and in the direction. So:

example_cond_num_A3.png

Now we realize that is a rotation by so:

example_cond_num_A2.png

Finally we realize that is stretching by in the direction and in the direction. So:

example_cond_num_A1.png

Looking at the picture above we can tell that the condition number is .