Solve:

1
Sec7.5 Multiple Eigenvalue Solution
Lin. indep eigenvectors
One single eigenvector
Solve
2
Repeated real Eigenvalues
One single eigenvector
3
Repeated real Eigenvalues
Solve
4
Repeated real Eigenvalues
Solve
5
Repeated real Eigenvalues
Solve
6
Homog Linear System
2X2 system
2 complex
2 real distinct
2 real repeated
Chain G-eigvec
2 lin indep eig-vec
3X3 system
1 real 2 complex
3 real distinct
2 real repeated 1 real
3 real repeated
2 lin indep eig-vec
3 lin indep eig-vec
2 lin indep eig-vec
3 lin indep eig-vec
1 lin indep eig-vec
7
Repeated real Eigenvalues
DEF
8
Repeated real Eigenvalues
rank 2 generalized eigenvector
rank 3 generalized eigenvector
DEF
A rank r generalized eigenvctor associated with
is a vector v such that
9
Repeated real Eigenvalues
10
Repeated real Eigenvalues
DEF
A length k chain of generalized eigenvectors
based on the eigenvector is a set of
of k generalized
eigenvectors such that
11
Jordan Block
Example
Example
Definition

• Find charc. Equ.
• Find all eigenvalues
• How many free variables
• How many lin. Indep eigvct
• defect

Jordan block with eigenvalue

• Find charc. Equ.
• Find all eigenvalues
• How many free variables
• How many lin. Indep eigvct
• defect

Chain of generalized eigenvectors
Examples
12
Jordan Normal Form
Exmples
Definition
Where each submatix is a jordan block of
the form

• Find eigenvalues
• multiplicity
• How maany lin. Indep eigenvectors
• How many chain and length

13
Jordan Normal Form
Theorem 1
Any nxn matrix A is similar to a Jordan
normal form matrix
Theorem 1
Let A be nxn matrix
there exits an invertable Q such that
where J is in Jordan normal form
Find the Jordan form
Find the Jordan form
14
Jordan Normal Form
Theorem 1
Let A be nxn matrix
there exits an invertable Q such that
where J is in Jordan normal form
If all generalized eigenvectors are arranged as
column vectors in proper order corresponding to
the appearance of the Jordan blocks in (), the
results is the matrix Q
Let A be 5x5 matrix
15
Another method to compute Generalized
eigenvectors
Compute
Solve
Compute
Chain of generalized eigenvectors
Find all generalized eigenvectors
Chain of generalized eigenvectors
16
Another method to compute Generalized
eigenvectors
Compute
Find all generalized eigenvectors
Solve
Compute
1 lin indep eigenvector
Length of chain 3
Chain of generalized eigenvectors