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Margot Gerritsen (Stanford Univ) on the beauty of Linear Algebra

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http://www.scaruffi.com/leonardo/sep2013.html

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Margot Gerritsen (Stanford Univ) on the beauty of Linear Algebra

  1. 1. Visualizing matrices with special thanks to Tim Davis and his beautiful matrix collection, see http://www.cise.ufl.edu/research/sparse/matrices/ Margot Gerritsen ICME, Stanford margot.gerritsen@stanford.edu
  2. 2. Matrices are everywhere
  3. 3. Visualizing using spy plots
  4. 4. A small system of equations w + y = 1 x + y + z = 1 w + x + y + z = 1 x + y + z = 1
  5. 5. Written as a matrix-vector equation 1 1 w 1 1 1 1 x 1 1 1 1 1 y 1 1 1 1 z 1 =
  6. 6. 4 1 1 Every nonzero becomes a dot This is a “spy plot”
  7. 7. Spy plots of two subdomains of the Stanford internet
  8. 8. Spy plot of a matrix occuring in oil reservoir modeling
  9. 9. Spy plots where size of element is reflected in color
  10. 10. Visualizing using graphs
  11. 11. Back to our matrix-vector equation 1 1 w 1 1 1 1 x 1 1 1 1 1 y 1 1 1 1 z =
  12. 12. We can draw it in different ways Which is most appealing?
  13. 13. But what if the matrix is bigger?
  14. 14. Alaric Hall, Uni of Leeds ÁlaFlekkssaga Bæringssaga Blómstrvallasaga DámustasagaokJóns DínussagaDrambláta Drauma-Jónssaga Flóressagakonungsoksonahans Gibbonssaga Hectorssaga HermannssagaokJarlmanns HringssagaokTryggva Jónssagaleikara Kirialaxsaga Klárisagakeisarasonar Konráðssagakeisarasonar Magnússagajarls Mírmannssaga Nitidasaga Rémundarsagakeisarasonar SálussagaokNikanors Samsonssagafagra Sigrgarðssagafrœkna SigrgarðssagaokValbrands Sigurðarsagafóts Sigurðarsagaturnara Sigurðarsagaþögla Tiódelssaga TristramssagaokÍsoddar Valdimarssaga ViktorssagaokBlávus Vilhjálmssagasjóðs Vilmundarsagaviðutan ÞjalarJónssaga Alexanderssaga AmícussagaokAmílíus Bevissaga Bretasögur ElissagaokRósamundu Erexsaga FlóressagaokBlankifúr Flóventssaga Ívenssaga Karlamagnússaga Möttulssaga Parcevalssaga Partalópasaga Strengleikar TristramssagaokÍsöndar Trójumannasaga Valvensþáttr
  15. 15. Here’s an idea: Give each node an electrical charge Make each edge a spring Drop the whole system on the floor and let it “wobble” until it finds its optimal (minimum energy) state
  16. 16. Financial portfolio optimization
  17. 17. Hessian matrix from a quadratic programming problem
  18. 18. Frequency-domain circuit simulation
  19. 19. Linear programming problem
  20. 20. Computational fluid dynamics: shallow-water equations
  21. 21. Linear programming problem
  22. 22. Social network: people and the web pages they like
  23. 23. LCSH Galaxy
  24. 24. Visualizing matrices with special thanks to Tim Davis and his beautiful matrix collection, see http://www.cise.ufl.edu/research/sparse/matrices/ Margot Gerritsen ICME, Stanford margot.gerritsen@stanford.edu

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