MAT 2401Linear Algebra

3.4 Introduction to Eigenvalues

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HW Written Homework

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

Notations Pay attention to the notations.

They will be confusing – numbers or vector?

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.

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

Example Preview – Population Modeling (Leslie Matrix) We can model the population from

year to year by the matrix equation1

1

1

n h a n

n h a n

n n

h B B ha S S a

x Ax

1 0

2 1

3 2

1n n

x Axx Axx Ax

x Ax

Example Preview – Population Modeling (Leslie Matrix) Stable proportion of population in

the age groups1

1

1

n h a n

n h a n

n n

h B B ha S S a

x Ax

1n nx x

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

1

1

1

n h a n

n h a n

n n

h B B ha S S a

x Ax

1n nx x

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

Example 1 (Eigenvector Given)

Find the eigenvalue of A if the eigenvector is

(a) (b)

1 42 3

A

1

11

x

2

21

x

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

x=Ax

Example 2Find the eigenvalues and

eigenvectors of 1 42 3

A

Remarks Eigenvalue and eigenvector always

come in pairs. Eigenvectors are unique up to

scalar multiple.

Example 3Find the eigenvalues and

eigenvectors of 1 2 21 2 11 1 0

A

Remarks1. det(I-A)=0 is called the

_________________ of A.2. It is a polynomial equation of

degree ___.

Visual Summary

2 2 5 4 6 8

Linear System

x y zx y zx y z

1 1 1 2 2 5

4 6 8

Agumented Matrix

1 0 2 10 1 3 20 0 0 0

# of Solutions

.23 ,

Parametric SolutionsLet z t

x ty t t Rz t