1. Complex Eigenvalues and Phase Portraits

2. Fundamental Set of Solutions For Linear System of ODEs With Eigenvalues and Eigenvectors and The General Solution Takes The Form

3. Not All Matrices Have Real Eigenvalues/Eigenvectors Has Eigenvalues and Eigenvectors and, Has The General Solution

4. Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Lots of Complex Numbers We want a Real General Solution

5. Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Lots of Complex Numbers We want a Real General Solution Recall Euler’s Formula

6. Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Or Just a Constant

7. Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Or Just a Constant

8. Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Or Just a Constant

9. Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Or Just a Constant

10. Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Or

11. Some Things To Notice Has Eigenvalues and Eigenvectors and, Are Complex Conjugates

12. Some Things To Notice Has Eigenvalues and Eigenvectors and, Are Complex Conjugates

13. This Will Always Hold If Has Complex Eigenvalues and Eigenvectors then and So In Practice, Only Need To Find and

14. This Will Always Hold then and So In Practice, Only Need To Find and If This Has Complex Eigenvalues and Eigenvectors

15. General Solution The General Solution Is If This Has Eigenvalue and Eigenvector

16. General Solution The General Solution Is If This Has Eigenvalue and Eigenvector

17. Questions?

18. Phase Portraits Remember Direction Fields: t y 2.1 0.5 ç ç 1.0 ç ç 1.5 ç ç ç ç 3.1 4.1 ç ç ç ç ç ç ç ç ç ç ç ç

19. We Can Draw Direction Fields For 2D Systems

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34. A Plot With Many Solutions Is Called a “Phase Portrait”

35. Phase Portraits Give Us An Idea of How Solution Behaves

36. Summary Can Use Euler’s Formula To Get General Solutions To Systems of Equations With Complex Eigenvalues Can Use Phase Portraits To Examine The Behavior Of Different Systems

37. Questions?