Scientific Computing Using Python - PHYS:4905
Homework #15
Due 11/12/2019


Name _______________________________________________

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


1. (15) Find the eigenvalues of the matrix below, b* is the complex conjugate of b.  Are the eigenvalues always real?  Note that this is a generalization of a symmetric matrix called a Hermitian matrix.  In quantum mechanics, linear operators that correspond to physical observables are Hermitian operators.


$$M = \begin{pmatrix} a & b \\ b^* & d \\ \end{pmatrix}$$


2. Let L be the linear transformation L: $ℝ^3 → ℝ^3$ given by

$$L \, \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} = \begin{pmatrix} 2x+y+3z \\ x + 2y +z \\ 3x + y +2z \end{pmatrix}$$

A. (6) Write L as a matrix M in the standard basis and find the eigenvalues of L.
B. (20) Find the eigenvectors of L in the standard basis.
C. (3) Calculate the dot product of your two eigenvectors.


3. (25) A discrete dynamical system, can be described by a vector v_n that gives the state of the system at a particular time t_n = n Δt,   and a matrix that describes how the system evolves in discrete time jumps of Δt,

$$v_{n+1} = M v_n$$
Fixed points are states of the system that do not evolve with time, i.e. if v_n is a fixed point, then v_n+1 = v_n ,  v_n+2 = v_n+1 , ...

Invariant curves are states of the system that evolve along the same direction.  If v_n is on an invariant curve, then v_n+1 = a_n v_n ,  v_n+2 = a_n+1 v_n+1 , ... where the a_n's are scalars.

Consider a discrete dynamical system described by the matrix  $M = \begin{pmatrix} 3 & 2 \\ 2 & 3 \end{pmatrix}$.  Find any fixed points and invariant curves of the dynamical system.