Department

Mathematics

Course Code

MATA23H3

Professor

Sophie Chrysostomou

Eigenvalues and Eigenvectors

Deﬁnition: Let Abeann×nmatrix. Ascalarλis aneigenvalue of Aif thereis a

nonzero vector v∈Rn,suchthat Av=λv.In this case, vis called aneigenvectorof A

corresponding to the eigenvalue λ.

Howto Find the Eigenvaluesof aSquareMatrix

Deﬁnition: Let Abeann×nmatrix. The characteristic polynomial ofAis given

byp(λ)=|A−λI|.If λis aneigenvalue ofA,then theset

Eλ={x

Ax =λx}

is called the eigenspace ofλ.It contains the zero vectorand all the eigenvectors ofAcorre-

sponding to λ.

Note: Eλ=nullspace(A−λI ).

1c

2011 bySophie Chrysostomou

www.notesolution.com

Example: Let A=

2 0 0

1 2 −1

1 3 −2

1. Find the characteristic polynomial of A.

2. Find all of the eigenvalues of A.

3. Foreacheigenvalue λof A,ﬁnd its eingespace Eλ.

2c

2011 bySophie Chrysostomou

www.notesolution.com

Example: Let A=

1 1 −3

2 0 6

1−1 5

1. Find the characteristic polynomial of A.

2. Find all of the eigenvalues of A.

3. Foreacheigenvalue λof A,ﬁnd its eingespace Eλ.

3c

2011 bySophie Chrysostomou

www.notesolution.com

Over 90% improved by at least one letter grade.

OneClass has been such a huge help in my studies at UofT especially since I am a transfer student. OneClass is the study buddy I never had before and definitely gives me the extra push to get from a B to an A!

Leah — University of Toronto

Balancing social life With academics can be difficult, that is why I'm so glad that OneClass is out there where I can find the top notes for all of my classes. Now I can be the all-star student I want to be.

Saarim — University of Michigan

As a college student living on a college budget, I love how easy it is to earn gift cards just by submitting my notes.

Jenna — University of Wisconsin

OneClass has allowed me to catch up with my most difficult course! #lifesaver

Anne — University of California

Join OneClass

Access over 10 million pages of study

documents for 1.3 million courses.

Sign up

Join to view

OR

By registering, I agree to the
Terms
and
Privacy Policies

Already have an account?
Log in

Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.