Octave for Chapter 8
Examples in this section provide sample Octave code for computing eigenvalues. You can access our code through the link at the bottom of each template. Feel free to modify the code and experiment to learn more!
You can write your own code using Octave software or online Octave cells. To access Octave cells online, go to the Sage Math Cell Webpage, select OCTAVE as the language, enter your code, and press EVALUATE.
To ”save” or share your online code, click on the Share button, select Permalink, then copy the address directly from the browser window. You can store this link to access your work later or share this link with others. You will need to get a new Permalink every time you modify the code.
Octave Tutorial
Find eigenvalues and the corresponding eigenvectors of the given matrix
.
% Define A
A=[5 0 0;
1 2 -1;
1 3 -2];
% Find eigenvalues of A
eigen_A=eig(A)
% Find eigenvalues and eigenvectors of A
[V,D]=eig(A)
% D is a diagonal matrix whose diagonal entries are the eigenvalues of A
% Columns of V are the eigenvectors of A
We can find the characteristic polynomial of the given matrix as follows.
% Define A
A=[5 0 0;
1 2 -1;
1 3 -2];
% Find the coefficients of the char poly of A
char_poly = poly(A)
The output looks like this:
char_poly =
1 -5 -1 5
The output is a vector of coefficients, starting with the leading coefficient. In this case, we see that the characteristic polynomial is .
In this text, we find the characteristic polynomial by computing . Octave,
as well as some other textbooks, finds the characteristic polynomial based
on . Note that setting these expressions equal to zero to find eigenvalues
produces the same result. However, the coefficients of the characteristic
polynomials from the two approaches are negatives of each other. If you were
to do this problem by hand, using the method in this text, you would get
.
In Example ex:eigsrotation, you found that the rotation matrix has complex eigenvalues. Use
the eig command in Octave to compute the eigenvalues and eigenvectors of
.
Here is the input code:
% Define A
A=[sqrt(2)/2 -sqrt(2)/2;
sqrt(2)/2 sqrt(2)/2];
% Find eigenvalues of A
eigen=eig(A)
% Find eigenvalues and eigenvectors of A
[V,D]=eig(A)
% D is a diagonal matrix whose diagonal entries are the eigenvalues of A
% Columns of V are the eigenvectors of A
This is what we get for the output:
eigen =
(0.707107,0.707107)
(0.707107,-0.707107)
V =
(0,0.707107) (0,-0.707107)
(0.707107,0) (0.707107,-0)
D =
(0.707107,0.707107) (0,0)
(0,0) (0.707107,-0.707107)
Each complex number in the output is presented using its real and imaginary parts as and coordinates of a point. For example, the second eigenvalue, written as , is the complex number . Compare this to the eigenvalue in Example ex:eigsrotation. The corresponding eigenvector is the second column of the matrix : . This should be interpreted as . Compare this to the eigenvector in Example ex:eigsrotation. There is a discrepancy! Did we make a mistake? How do we reconcile the two eigenvectors?
Octave Exercises
Let .
- (a)
- Use Octave to find the eigenvalues of .
- (b)
- Find the eigenvalues of . How do they compare?
- (c)
- Formulate and prove a conjecture about the relationship between the eigenvalues of and the eigenvalues of , where is a constant.
- (d)
- How do the eigenvectors of compare to the eigenvectors of ? Prove your claim.
Let
- (a)
- Find the eigenvalues of using Octave. List the eigenvalues below in increasing order.
- (b)
- By hand, find the corresponding eigenvectors of . List them below.
- (c)
- Use Octave to find the eigenvectors of . Reconcile your answers from the previous part with the answers you got from Octave.
Use Octave to find the eigenvalues of . Interpret the numbers you are seeing.
Interpret your results geometrically.
Let
When we use Octave to find the eigenvalues of , we get the following printout.
ans =
16.1168
-1.11684
-1.30368e-15
Should we interpret the last eigenvalue as , or as a very small non-zero number? How can you be sure of your answer in this particular case?
Modify the code for the Power Method given in The Power Method and the
Dominant Eigenvalue to approximate the dominant eigenvalue of . Experiment with
the number of iterations. Compare your answer to the answer you get when using the
eig function.
2024-10-01 02:16:32