1. Jordan Block
Under what conditions a given matrix is diagonalizable ???
Therorem 1:
REMARK:
Not all nxn matrices are diagonalizable
A similar to (close to diagonal matrix)

2. Jordan Block Example Example Definition: Jordan block with eigenvalue
Find charc. Equ.
Find all eigenvalues
How many free variables
How many lin. Indep eigvct
Jordan block with eigenvalue
Find charc. Equ.
Find all eigenvalues
How many free variables
How many lin. Indep eigvct
of size k
Examples

3. Jordan Normal Form Exmples: Definition: is in Jordan normal form
Where each submatix is a jordan block of the form
Find eigenvalues
multiplicity
How maany lin. Indep eigenvectors
How many chain and length
Note: s = # lin.indep eigvectors

4. 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

5. Jordan Normal Form
Find the Jordan form
Find the Jordan form

6. Repeated real Eigenvalues
DEF

7. 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

8. Repeated real Eigenvalues

9. 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

10. 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

11. 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