PHYCS 3730/6720 Lab Exercise

Reading and references: Here we practice solving eigenvalue problems using Maple and Python/numpy. The answer file is Mylab11.txt. You may use Matlab instead of Maple if you like.

Exercise 1.

Use Python in interactive mode and use the numpy.linalg package to find the eigenvalues and eigenvectors of the matrix
   9   3   3
   3   2   4
   3   4   2
Copy your Python session (with the results) to the answer file.

Exercise 2.

To check your understanding, use Python (interactive mode) to verify that the eigenvector for the eigenvalue 12 satifies the eigenvalue equation: matrix times eigenvector equals eigenvalue times eigenvector. Copy the new parts of your Python session to the answer file.

Exercise 3.

Use Maple (or Matlab) to find the eigenvalues and eigenvectors of the same matrix.

Remember to use with(linalg). The functions eigenvals and eigenvectors are what you are looking for. Be sure you understand the output from the eigenvectors function. Copy your Maple session to your answer file.

Exercise 4.

You may have noticed that the Maple (or the Matlab) and numpy eigenvectors are not the same. This happens because the conventions for normalization may be different. Numpy normalizes the vectors to unit norm. Maple uses a different normalization. To see that is is only a matter of normalization, for each of the three eigenvectors, find the constant that, when it multiplies the Maple eigenvector, the resulting product matches the Python eigenvector. Do this by copying the Maple eigenvectors to Python (one-by-one) and using Python to figure out the normalization constant. In your answer file, for each eigenvector, show the Maple eigenvector, the constant, the result of multiplying by the constant, and the Python eigenvector for the sake of comparison.

Copy the relevant parts of your Python session to the answer file.