Summary linear algebra (notes, exercises, correction)

CHAPTER 5 EIGENVALUES AND EIGENVECTORS

5.1 Eigenvectors and Eigenvalues


• Eigenvector and Eigenvalue

A scalar λ is called an eigenvalue of A if there is a nontrivial solution x of
Ax = λx. Such an x is called an eigenvector corresponding to λ.


     
1 6 6 3
• Example Let A =  , u =  , and v =  . Are u and v
5 2 −5 −2
eigenvectors of A?




 
1 6
• Example Let A =  . Show that 7 is an eigenvalue of A and find the
5 2
corresponding eigenvectors.




1

,2

• The set of all solution of (A−λI)x = 0 is just the null space of the matrix A−λI.
So, this set is a subspace and is called the eigenspace of A corresponding to
λ.
 
4 −1 6
 
• Example Let A =  2 1 6 . An eigenvalue of A is 2. Find a basis for
 
 
2 −1 8
the corresponding eigenspace.




Exercises §5.1, 1,3,5,7,9,11,13,15,17.

, 3

5.2 The Characteristic Equation
 
2 3
• Example Find the eigenvalues of A =  .
3 −6




• The Characteristic Equation

A scalar λ is an eigenvalue of an n × n matrix A if and only if λ satisfies the
characteristic equation
det(A − λI) = 0

, 4

• Example Find the eigenvalues and eigenvectors of the matrix
 
 5 −2 6 −1 
 
 0 3 −8 0 
A=
 

0 0 5 4
 
 
0 0 0 1