1. 3.4 Introduction to Eigenvalues
MAT 2401 Linear Algebra
3.4 Introduction to Eigenvalues

2. HW
Written Homework

3. Overview Eigenvalues are used in a variety of real life applications.
Eigenvalues are central to many theories for applicable mathematics.
Eigenvalues is one of the most important topics in elementary linear algebra.
More to come in section 7

4. Notations
Pay attention to the notations. They will be confusing – numbers or vector?

5. Example Preview – Population Modeling (Leslie Matrix)
Suppose we are interested in the population of a certain type of bird in a forest area.
We can divide the population in two age groups – hatchlings (age<1) and adults.

6. Example Preview – Population Modeling (Leslie Matrix)
Suppose we can estimate the following parameters:
Birth rate from hatchlings Bh
Birth rate from adults Ba
Survival rate of hatchlings Sh
Survival rate of adults Sa

7. Example Preview – Population Modeling (Leslie Matrix)
We can model the population from year to year by the matrix equation

8. Example Preview – Population Modeling (Leslie Matrix)
Stable proportion of population in the age groups

9. Example Preview – Population Modeling (Leslie Matrix)
Q: How to find ?
A: From the relationship: Ax= x.

10. Eigenvalues and Eigenvectors
Let A be a nxn matrix,  a scalar, and x a non-zero nx1 column vector.
 and x are called an eigenvalue and eigenvector of A respectively if
Ax= x

11. Example 1 (Eigenvector Given)
Find the eigenvalue of A if the eigenvector is (a) (b)

12. How to Find the Eigenvalues and Eigenvector?
Recall: Theorem from 3.3
A is invertible if and only if det(A)≠0
Equivalently
A is singular if and only if det(A)=0
Now, …
x=Ax

13. Example 2
Find the eigenvalues and eigenvectors of

14. Remarks Eigenvalue and eigenvector always come in pairs.
Eigenvectors are unique up to scalar multiple.

15. Example 3
Find the eigenvalues and eigenvectors of

16. Remarks
1. det(I-A)=0 is called the _________________ of A. 2. It is a polynomial equation of degree ___.

17. Visual Summary