Section 5.1 Eigenvectors and Eigenvalues

1. Section 5.1Eigenvectors and Eigenvalues

2. Eigenvectors and Eigenvalues Useful throughout pure and applied mathematics. Used to study difference equations and continuous dynamical systems. Provide critical information in engineering design Arise naturally in such fields as physics and chemistry. Used in statistics to analyze multicollinearity

3. Example: Let Consider Av for some v

4. Definition: The Eigenvector of Anxnis a nonzero vector x such that Ax=λx for some scalar λ. λ is called an Eigenvalue of A

5. Statistics (multicollinearity) Where y is the dependent response vector and the x’s are the independent explanatory vectors The β’s are least squares regression coefficients εi are errors We desire linear independence between x vectors Can use Eigen analysis to determine

6. From Definition: Ax = λx = λIx Ax – λIx = 0 (A – λI)x = 0 Observations: λ is an eigenvalue of A iff (A – λI)x=0 has non-trivial solutions A – λI is not invertible IMT all false The set {xεRn: (A – λI)x=0} is the nullspace of (A – λI)x=0, A a subspace of Rn The set of all solutions is called the eigenspace of A corresponding to λ

7. Example Show that 2 is an eigenvalue of and find the corresponding eigenvectors.

8. Comments: Warning: The method just used (row reduction) to find eigenvectors cannot be used to find eigenvalues. Note: The set of all solutions to (A-λI)x =0 is called the eigenspace of A corresponding to λ.

9. Example Let An eigenvalue of A is λ=3. Find a basis for the corresponding eigenspace.

10. Theorem The eigenvalues of a triangular matrix are the entries on its main diagonal.

11. Proof of 3x3 case Let So (A-λI) =

12. Proof of 3x3 case By definition λ is an eigenvalueiff (A-λI)x=0 has non-trivial solutions so a free variable must exist. This occurs when a11=λ or a22=λ or a33=λ

13. Example Consider the lower triangular matrix below. λ= 4 or 0 or -3

14. Addtion to IMT Anxn is invertible iff The number 0 is not an eigenvalue det A≠0 (not sure why author waits until not to add this)

15. Theorem If eigenvectors have distinct eigenvalues then the eigenvectors are linearly independent This can be proven by the IMT

16. Section 5.2The Characteristic Equation

17. Finding Eigenvalues We know (A-λI)x=0 must have non-trivial solutions and x is non-zero. That is free variables exist. So (A-λI) is not invertible by the IMT Therefore det(A-λI)=0 by IMT

18. Characteristic Equation det(A-λI)=0 Solve to find eigenvalues Note: det(A-λI) is the characteristic polynomial.

19. Previous Example: Let find eigenvalues

20. Example Find Eigenvalues

21. Example Find characteristic polynomial and eigenvalues

22. Example Find the characteristic polynomial Find all eigenvalues Find multiplicity of each eigenvalue

23. Recap λ is an eigenvalue of A if (A-λI)x=0 has non-trivial solutions (free variables exist). Eigenvectors (eigenspace )are found by row reducing (A-λI)x=0. Eigenvalues are found by solving det(A-λI)=0.

24. Section 5.3Diagonalization

25. Diagonalization The goal here is to develop a useful factorization A=PDP-1, when A is nxn. We can use this to compute Ak quickly for large k. The matrix D is a diagonal matrix (i.e. entries off the main diagonal are all zeros).

26. Example Find a formula for Akgiven A=PDP-1 &

27. Diagonalizable Matrix “A” is diagonalizable if A=PDP-1 where P is invertible and D is a diagonal matrix. Note: AP=PD

28. When is a matrix diagonalizable? Let’s examine eigenvalues and eigenvectors of A

29. The Diagonalization Theorem IfAnxn & has n linearly independent eigenvectors. Then A=PDP-1 Columns of P are eigenvectors Diagonals of D are eigenvalues.

30. Example Diagonalize We need to find P & D

31. Theorem IfAnxn has n distinct eigenvaluesthen A is diagonalizable.

32. Example Diagonalize

33. Example Diagonalize