Understanding Eigenvalues and Characteristic Polynomials

 
Characteristic
Polynomial
 
Hung-yi Lee
 
Outline
 
Last lecture:
Given eigenvalues, we know how to find
eigenvectors or eigenspaces
Check eigenvalues
This lecture: How to find eigenvalues?
 
Reference: Textbook 5.2
Looking for Eigenvalues
Characteristic Polynomial
Eigenvalues are the roots of characteristic
polynomial or solutions of characteristic equation.
 
Characteristic polynomial of A
 
Characteristic equation of A
 
A is the standard matrix of linear operator T
 
linear operator T
 
linear operator T
Looking for Eigenvalues
Example 1: Find the eigenvalues of
 
The eigenvalues of 
A
 are -3 or 5.
 
=0
Looking for Eigenvalues
Example 1: Find the eigenvalues of
The eigenvalues of 
A
 are -3 or 5.
 
Eigenspace of -3
 
Eigenspace of 5
 
find the solution
 
find the solution
Looking for Eigenvalues
Example 2: find the eigenvalues of linear operator
 
standard
matrix
Looking for Eigenvalues
Example 3: linear operator on 
R
2
 that rotates a
vector by 90
 
No eigenvalues, no eigenvectors
Characteristic Polynomial
 
In general, a matrix A and RREF of A have different
characteristic polynomials.
Similar matrices have the same characteristic
polynomials
 
Different Eigenvalues
 
The same Eigenvalues
Characteristic Polynomial
 
Question: What is the order of the characteristic polynomial
of an 
n
n
 matrix 
A
?
The 
characteristic polynomial of an 
n
n
 matrix is indeed
a polynomial with degree 
n
Consider
 
det(
A
 
 
tI
n
)
Question: What is the number of eigenvalues of an 
n
n
matrix 
A
?
Fact: An 
n
 x 
n
 matrix 
A
 have less than or equal to 
n
eigenvalues
Consider complex roots and multiple roots
Characteristic Polynomial
If nxn matrix A has 
n
 eigenvalues (
including
multiple roots)
Sum of  n
eigenvalues
Product of  n
eigenvalues
Trace of A
Determinant of
A
 
=
 
=
 
Eigenvalues:
-3, 5
 
Example
Characteristic Polynomial
The eigenvalues of an upper triangular matrix are
its diagonal entries.
 
The determinant of an upper triangular matrix
is the product of its diagonal entries.
Characteristic Polynomial
v.s. Eigenspace
Characteristic polynomial of A is
Factorization
 
Eigenvalue:
 
Eigenspace:
 
(dimension)
multiplicity
Characteristic Polynomial
v.s. Eigenspace
Example 1:
 
characteristic polynomials:
 
(
t
 + 1)
2
(
t
 
 3)
 
Eigenvalue -1
 
Eigenvalue 3
 
Multiplicity of “-1” is 2
 
Multiplicity of “3” is 1
 
Dim of eigenspace is 1 or 2
 
Dim of eigenspace must be 1
 
Dim = 2
Characteristic Polynomial
v.s. Eigenspace
Example 2:
 
characteristic polynomials:
 
(
t
 + 1)
 
(
t
 
 3)
2
 
Eigenvalue -1
 
Eigenvalue 3
 
Multiplicity of “-1” is 1
 
Multiplicity of “3” is 2
 
Dim of eigenspace is 1 or 2
 
Dim of eigenspace must be 1
 
Dim = 2
Characteristic Polynomial
v.s. Eigenspace
Example 3:
 
characteristic polynomials:
 
(
t
 + 1)
 
(
t
 
 3)
2
 
Eigenvalue -1
 
Eigenvalue 3
 
Multiplicity of “-1” is 1
 
Multiplicity of “3” is 2
 
Dim of eigenspace is 1 or 2
 
Dim of eigenspace must be 1
 
Dim = 1
Characteristic
polynomial
Eigenvalues
Eigenspaces
 
(
t
 + 1)
2
(
t
 
 3)
 
(
t
 + 1)
 
(
t
 
 3)
2
 
(
t
 + 1)
 
(
t
 
 3)
2
 
-1
 
3
 
-1
 
3
 
-1
 
3
 
2
 
1
 
1
 
2
 
1
 
1
Summary
Characteristic polynomial of A is
Factorization
 
Eigenvalue:
 
Eigenspace:
 
(dimension)
multiplicity
 
Homework
 
Slide Note
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Unravel the mystery of eigenvalues and characteristic polynomials through detailed lectures and examples by Hung-yi Lee. Learn how to find eigenvalues, eigenvectors, and eigenspaces, and explore the roots of characteristic polynomials to solve characteristic equations. Dive into examples to discover eigenvalues of linear operators and matrices, and understand the significance of eigenspaces in different scenarios.


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  1. Characteristic Polynomial Hung-yi Lee

  2. Outline Last lecture: Given eigenvalues, we know how to find eigenvectors or eigenspaces Check eigenvalues This lecture: How to find eigenvalues? Reference: Textbook 5.2

  3. Looking for Eigenvalues A scalar ? is an eigenvalue of A ?? = ?? Existing ? 0 such that ?? ?? = 0 Existing ? 0 such that Existing ? 0 such that ? ???? = 0 ? ???? = 0 has multiple solution The columns of ? ??? are independent Dependent ? ??? is not invertible Rank ? ??? < n ??? ? ??? = 0

  4. Characteristic Polynomial ??? ? ??? = 0 A scalar ? is an eigenvalue of A A is the standard matrix of linear operator T ??? ? ???: Characteristic polynomial of A linear operator T ??? ? ??? = 0: Characteristic equation of A linear operator T Eigenvalues are the roots of characteristic polynomial or solutions of characteristic equation.

  5. Looking for Eigenvalues Example 1: Find the eigenvalues of ??? ? ??? = 0 A scalar ? is an eigenvalue of A =0 t = -3 or 5 The eigenvalues of A are -3 or 5.

  6. Looking for Eigenvalues Example 1: Find the eigenvalues of The eigenvalues of A are -3 or 5. Eigenspace of -3 ?? = 3? ? + 3? ? = 0 find the solution Eigenspace of 5 ?? = 5? ? 5? ? = 0 find the solution

  7. Looking for Eigenvalues Example 2: find the eigenvalues of linear operator standard matrix ??? ? ??? = 0 A scalar ? is an eigenvalue of A 1 ? 2 0 0 0 ? ???= 1 ? 0 1 1 ? ??? ? ??? = 1 ?3

  8. Looking for Eigenvalues Example 3: linear operator on R2that rotates a vector by 90 ??? ? ??? = 0 A scalar ? is an eigenvalue of A standard matrix of the 90 -rotation: No eigenvalues, no eigenvectors

  9. Characteristic Polynomial In general, a matrix A and RREF of A have different characteristic polynomials. Similar matrices have the same characteristic polynomials The same Eigenvalues Different Eigenvalues ? = ? 1?? = ??? ? 1?? ? 1?? ? ??? ? ?? = ??? ? 1? ?? ?? = ??? ? 1??? ? ?? ??? ? 1 = ??? ? ?? ??? ? = ??? ? ?? ??? ?

  10. Characteristic Polynomial Question: What is the order of the characteristic polynomial of an n n matrix A? The characteristic polynomial of an n n matrix is indeed a polynomial with degree n Consider det(A tIn) Question: What is the number of eigenvalues of an n n matrix A? Fact: An n x n matrix A have less than or equal to n eigenvalues Consider complex roots and multiple roots

  11. Characteristic Polynomial If nxn matrix A has n eigenvalues (including multiple roots) Sum of n eigenvalues = Trace of A Product of n eigenvalues Determinant of A = Example Eigenvalues: -3, 5

  12. Characteristic Polynomial The eigenvalues of an upper triangular matrix are its diagonal entries. Characteristic Polynomial: ? ? 0 0 ? 0 0 ? 0 ? ??? ? ? 0 ? ? = ? ? ? ? ? ? The determinant of an upper triangular matrix is the product of its diagonal entries.

  13. Characteristic Polynomial v.s. Eigenspace Characteristic polynomial of A is ??? ? ??? Factorization multiplicity ?1? ?2 ?2 ? ?? ?? = ? ?1 ?1 ?2 ?? Eigenvalue: ?1 ?? Eigenspace: (dimension) ?2 ?? ?1 ?2

  14. Characteristic Polynomial v.s. Eigenspace Example 1: characteristic polynomials: (t + 1)2(t 3) Eigenvalue -1 Multiplicity of -1 is 2 Dim of eigenspace is 1 or 2 Dim = 2 Eigenvalue 3 Multiplicity of 3 is 1 Dim of eigenspace must be 1

  15. Characteristic Polynomial v.s. Eigenspace Example 2: characteristic polynomials: (t + 1)(t 3)2 Eigenvalue -1 Multiplicity of -1 is 1 Dim of eigenspace must be 1 Eigenvalue 3 Multiplicity of 3 is 2 Dim of eigenspace is 1 or 2 Dim = 2

  16. Characteristic Polynomial v.s. Eigenspace Example 3: characteristic polynomials: (t + 1)(t 3)2 Eigenvalue -1 Multiplicity of -1 is 1 Dim of eigenspace must be 1 Eigenvalue 3 Dim of eigenspace is 1 or 2 Multiplicity of 3 is 2 Dim = 1

  17. Characteristic polynomial Eigenvalues Eigenspaces -1 2 (t + 1)2(t 3) 3 1 -1 1 (t + 1)(t 3)2 3 2 1 -1 (t + 1)(t 3)2 1 3

  18. Summary Characteristic polynomial of A is ??? ? ??? Factorization multiplicity ?1? ?2 ?2 ? ?? ?? = ? ?1 ?1 ?2 ?? Eigenvalue: ?1 ?? Eigenspace: (dimension) ?2 ?? ?1 ?2

  19. Homework

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