Eigen Values and Eigen Vectors
Question 1: How to find Eigenvalues and eigen vectors by hand?
M=
What are eigenvalues and eigenvectors?
Special combination of a scalar: and a vector that:
How to find it by hand?
Find Eigen Values
Find Eigen vectors
Let
Eigen vector is
There is always "infinity solutions of eigen vectors!!! (But in the same direction)"
Let
Can we diagonalize the matrix?
Let
Find !!!
First row + 2$\timesx_2$):
Quiz 1:
If we define
If we define , what will be if
(a) 0
(b) infinity
(c) 3
Quiz 2:
For , , what will be?
(a) 84635418
(b) 0
(c) 42317709
Complex Numbers
What will happen if eigenvalues are complex numbers? (Why I say "eigenvalues are" not "eigenvalue is?")
Quiz 3:
M =
eigenvalues are:
eigenvectors are:
If we define
What will happen if
(a)spiral out
(b)spiral in
(c)stable limit cycle
Why?
(This will not be tested, but good to know)
Any complex number can be written as
(Pythagorean theorem)
Some Example Questions
Question 2:
The Siberian tiger is an endangered subspecies of tiger that inhabits forests in Siberia and northern China. Set up a discrete-time model for a Siberian tiger population based on the assumptions below. If your model is linear, write down its matrix. If not, explain why not.
The population is divided into cubs (less than a year old), subadults (1–3 years old) and adults.
On average, adults have 1.5 cubs per year.
Bengal tiger cubs only remain cubs for one year. 52% of cubs die during this first year.
On average, 63% of subadults survive as subadults from one year to the next.
20% of subadults mature into adults each year
10% of adults die each year
We define three state variables
Cubs: , Subadults: , and Adult:
On average, adults have 1.5 cubs per year.
Bengal tiger cubs only remain cubs for one year. 52% of cubs die during this first year.
All cubs are not cubs the next year, and (100%-52%)= 48% of them become subadults.
On average, 63% of subadults survive as subadults from one year to the next.
20% of subadults mature into adults each year
10% of adults die each year
Question 2:
The following discrete-time matrix model describes the population of alligators in Florida, where the population has been divided into three life stages: juveniles (J), early adults (E), and adults (A).
The eigenvalues of the matrix in this model are 0.67, 0.96, and 0.21. Regardless of the initial state, what will happen to the alligator population in the long run? (Be as specific as possible.)
Find the dominant eigenvector of this model
=
will solve.
eigenvector is