Lin. indep eigenvectors One single eigenvector Sec7.5: Multiple Eigenvalue Solution Lin. indep eigenvectors One single eigenvector Solve:

Repeated real Eigenvalues One single eigenvector

Repeated real Eigenvalues Solve:

Repeated real Eigenvalues Solve:

Repeated real Eigenvalues Solve:

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

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 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

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 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

Another method to compute: Generalized eigenvectors Solve: Compute: Chain of generalized eigenvectors Find all generalized eigenvectors: Chain of generalized eigenvectors

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