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)

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

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

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

Jordan Normal Form Find the Jordan form Find the Jordan form

Repeated real Eigenvalues DEF

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

Repeated real Eigenvalues

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

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

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