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Solve:

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.

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Solve:

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

  2. Repeated real Eigenvalues One single eigenvector

  3. Repeated real Eigenvalues Solve:

  4. Repeated real Eigenvalues Solve:

  5. Repeated real Eigenvalues Solve:

  6. 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

  7. Repeated real Eigenvalues DEF

  8. 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

  9. Repeated real Eigenvalues

  10. 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

  11. Jordan Block Example Example Definition: • 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

  12. 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

  13. 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 Find the Jordan form

  14. 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

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

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

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