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

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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