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Lecture

Eigenvalues, eigenvectors, and diagonalization

14 Pages
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Department
Mathematics
Course Code
MATA23H3
Professor
Sophie Chrysostomou

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Eigenvalues and Eigenvectors
Definition: Let Abeann×nmatrix. Ascalarλis aneigenvalue of Aif thereis a
nonzero vector vRn,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)=|AλI|.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(AλI ).
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
11 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

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Description
Eigenvalues and Eigenvectors Denition: Let A be an n n matrix. A scalar is an eigenvalue of A if there is a n nonzero vector v R , such that Av = v. In this case, v is called an eigenvector of A corresponding to the eigenvalue . How to Find the Eigenvalues of a Square Matrix Denition: Let A be an n n matrix. The characteristic polynomial of A is given by p() = A I. If is an eigenvalue of A, then the set E {x Ax = x} is called the eigenspace of . It contains the zero vector and all the eigenvectors of A corre- sponding to . Note: E =nullspace(A I ). 1 2011 by Sophie Chrysostomou www.notesolution.com 2 0 0 Example: Let A = 1 2 1 1 3 2 1. Find the characteristic polynomial of A. 2. Find all of the eigenvalues of A. 3. For each eigenvalue of A, nd its eingepace E . 2 2011 by Sophie Chrysostomou www.notesolution.com
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