Jordan Block Under what conditions a given matrix

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

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

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

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

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

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 5 x 5 matrix