Applied Geometrical Matrix Computations

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Outline. Geometrical Matrix ComputationsIllustration with 2x2 matrices:Excursions into eigenland (or why tangency and curvature matter!!)Where do matrix factorizations come from?Application to Color ScienceMatrix Animations. What exists? Maybe a survey?. Outline. Geometrical Matrix Computation

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Applied Geometrical Matrix Computations

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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

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