15. Eigenstuff (Concepts, No Computations)

Eigenvectors, eigenvalues, eigenspaces, eigenbases. Learn what they are and how - and why - they relate to diagonalization without being bogged down by computational algorithms... yet. This is the fifteenth in what will eventually be a complete sequence of supplementary videos for a linear algebra class that I teach based on my book 𝑇ℎ𝑒 𝐷𝑎𝑟𝑘 𝐴𝑟𝑡 𝑜𝑓 𝐿𝑖𝑛𝑒𝑎𝑟 𝐴𝑙𝑔𝑒𝑏𝑟𝑎. See my website https://bravernewmath.com for more information on my various books: 𝐹𝑢𝑙𝑙 𝐹𝑟𝑜𝑛𝑡𝑎𝑙 𝐶𝑎𝑙𝑐𝑢𝑙𝑢𝑠, 𝑇ℎ𝑒 𝐷𝑎𝑟𝑘 𝐴𝑟𝑡 𝑜𝑓 𝐿𝑖𝑛𝑒𝑎𝑟 𝐴𝑙𝑔𝑒𝑏𝑟𝑎, 𝑃𝑟𝑒𝑐𝑎𝑙𝑐𝑢𝑙𝑢𝑠 𝑀𝑎𝑑𝑒 𝐷𝑖𝑓𝑓𝑖𝑐𝑢𝑙𝑡, 𝐿𝑜𝑏𝑎𝑐ℎ𝑒𝑣𝑠𝑘𝑖 𝐼𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑒𝑑. The first three are available for sale as paperbacks at Amazon, and as pdfs at Lulu. (The Lobachevski book is available at Amazon and the American Mathematical Society) The animations at the end of the video were done using a "linear transformation visualizer" that you can find and play with at https://shad.io/MatVis/ . 0:00 Intro 1:29 Definitions - eigenvectors and eigenvalues 3:11 Geometric meaning - stretchin' out 6:44 The kernel as an eigenspace 11:18 Eigenspaces in general 19:38 Eigenbasis 26:07 A matrix rep. relative to an eigenbasis... 28:12 ...is a diagonal matrix 30:54 Example in R2 37:01 Diagonalizable maps 37:53 Examples of non-diagonalizable maps 40:40 Why don't we consider the zero vector an eigenvector? 43:50 Animations