Scientific Computing Using Python - PHYS:4905 - Fall 2018
Homework #14
Due 11/27/2018
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.
2. Let L be the linear transformation L: given
by
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 vn
that gives the state of the system at a particular time tn
= n Δt, and a matrix that describes how the system
evolves in discrete time jumps of Δt,
Fixed points are states of the system that do not evolve with
time, i.e. if vn is a fixed point, then vn+1
= vn , vn+2 =
vn+1 , ...
Invariant curves are states of the system that evolve along
the same direction. If vn is on an
invariant curve, then vn+1 = an
vn , vn+2 = an+1
vn+1 , ... where the an's
are scalars.
Consider a discrete dynamical system described by the matrix
. Find any
fixed points and invariant curves of the dynamical system.