1. Lin. indep eigenvectors One single eigenvector
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: Jordan block with eigenvalue
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: 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
Solve:
Compute:
Chain of generalized eigenvectors
Find all generalized eigenvectors:
Chain of generalized eigenvectors

16. Another method to compute: Generalized eigenvectors
Find all generalized eigenvectors:
Solve:
Compute:
1 lin indep
eigenvector
Length of chain =3
Chain of generalized eigenvectors