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

2. Example 1)Find charc. Equ. 2)Find all eigenvalues 3)How many free variables 4)How many lin. Indep eigvct Example 1)Find charc. Equ. 2)Find all eigenvalues 3)How many free variables 4)How many lin. Indep eigvct Definition: Jordan block with eigenvalue Examples Jordan Block of size k

3. Definition: Where each submatix is a jordan block of the form Jordan Normal Form Exmples: 1)Find eigenvalues 2)multiplicity 3)How maany lin. Indep eigenvectors 4)How many chain and length is in Jordan normal form Note: s = # lin.indep eigvectors

4. Theorem 1: Any nxn matrix A is similar to a Jordan normal form matrix Jordan Normal Form Theorem 1: Let A be nxn matrixthere exits an invertable Q such that: where J is in Jordan normal form Find the Jordan form

5. Jordan Normal 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. DEFA length k chain of generalized eigenvectors based on the eigenvector is a set of of k generalized eigenvectors such that

10. Example 1)Find charc. Equ. 2)Find all eigenvalues 3)How many free variables 4)How many lin. Indep eigvct 5)defect Example 1)Find charc. Equ. 2)Find all eigenvalues 3)How many free variables 4)How many lin. Indep eigvct 5)defect Definition: Jordan block with eigenvalue Chain of generalized eigenvectors Examples Jordan Block

11. Jordan Normal Form Theorem 1: Let A be nxn matrixthere 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