Scientific Computing Using Python - PHYS:4905
Homework #14
Due 11/5/2019


Name _______________________________________________

The answers to these questions must be hand written on paper and all work must be shown to receive full credit.


1. (10) Find the eigenvalues of the matrix below.

$$M = \begin{pmatrix} 2 & 1 & -1 \\ 1 & 2 & -1 \\ -1 & -1 & 2 \\ \end{pmatrix}$$

2. Let L be the linear transformation L: $ℝ^2 → ℝ^2$ given by

$L \, \begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} x+y \\ x + 3y \end{pmatrix}$

A. (6) Write L as a matrix M in the standard basis and find the eigenvalues of L.
B. (15) Find the eigenvectors of L in the standard basis.
C. (3) Find the change of basis matrix P to convert to the basis where L is diagonal.
D. (3) Calculate A = MP.
E. (3) Calculate B = P^-1A.  Is B a diagonal matrix with entries equal to the eigenvalues of L?
F. (3) Calculate the dot product of your two eigenvectors.


3. (15) Let V be the vector space of smooth (i.e. infinitely differentiable) functions f(x) that map the real numbers into the real numbers.  The second derivative $\frac{d^2}{dx^2}$ is then a linear operator that maps V into V.  What are the eigenfunctions of the second derivative operator and what are the corresponding eigenvalues?