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#### Presentation Transcript

**1. **Applied Geometrical Matrix Computations
Alan Edelman
Dept of Mathematics: MIT
MIT Laboratory for Computer Science

**2. **Outline Geometrical Matrix Computations
Illustration with 2x2 matrices:
Excursions into eigenland (or why tangency and curvature matter!!)
Where do matrix factorizations come from?
Application to Color Science
Matrix Animations

**3. **Outline Geometrical Matrix Computations
Illustration with 2x2 matrices:
Excursions into eigenland (or why tangency and curvature matter!!)
Where do matrix factorizations come from?
Application to Color Science
Matrix Animations

**4. **Geometrical Matrix Computations

**5. **Vector Space Diagrams

**6. **Outline Geometrical Matrix Computations
Illustration with 2x2 matrices:
Excursions into eigenland (or why tangency and curvature matter!!)
Where do matrix factorizations come from?
Application to Color Science
Matrix Animations

**7. **Eigenland (in 2d)

**8. **Eigenland (in 2d)

**9. **The Eigenvalue Map

**10. **The Eigenvalue Map

**11. **The Eigenvalue Map

**12. **Pseudospectra (Trefethen)

**13. **pseudospectra & geometry

**14. **Outline Geometrical Matrix Computations
Illustration with 2x2 matrices:
Excursions into eigenland (or why tangency and curvature matter!!)
Where do matrix factorizations come from?
Application to Color Science
Matrix Animations

**19. **Outline Geometrical Matrix Computations
Illustration with 2x2 matrices:
Excursions into eigenland (or why tangency and curvature matter!!)
Where do matrix factorizations come from?
Application to Color Science
Matrix Animations

**20. **Where do Matrix Factorizations Come From?

**21. **Semisimple group recipe

**22. **Matrix Factorizations Where can we look for new factorizations?
The Mathematics Literature
Lie Algebra: Cartan, Iwasawa, Bruhat
Representation Theory: Quivers
Nearness Problems
Applications
Engineering: A factorization is useful if someone can use it
Mathematics: The useful factorizations are characterized by an abstract criterion

**23. **Ideas to Generalize

**24. **Ideas to Generalize

**25. **Ideas to Generalize

**26. **Ideas to Generalize

**27. **Step 1:Cartan Decomposition

**28. **Step 1:Cartan Decomposition (U/O)

**29. **Step 1: Cartan Decomposition (On/Op X Oq )

**30. **Step 2:KAK Decomposition

**31. **Step 2:KAK Decomposition (U/Q)

**32. **Step 2:KAK Decomposition (On/Op X Oq )

**33. **Steps 3 and 4

**34. **Missing

**35. **Outline Geometrical Matrix Computations
Illustration with 2x2 matrices:
Excursions into eigenland (or why tangency and curvature matter!!)
Where do matrix factorizations come from?
Application to Color Science
Matrix Animations

**36. **Color Science: Light Spectra from film

**37. **Film Recording and measurements

**38. **The Data

**39. **SVD Basis = no physical meaning

**40. **The NNMF Basis = primary colors

**41. **Outline Geometrical Matrix Computations
Illustration with 2x2 matrices:
Excursions into eigenland (or why tangency and curvature matter!!)
Where do matrix factorizations come from?
Application to Color Science
Matrix Animations

**42. **Singular 2x2 Matrices (by svd)

**45. **Challenges