16. Eigendecomposition (Concepts, not Computations)

Diagonalization and eigendecomposition: seven-syllable nonsense words? Hardly. They are essential tools for making linear algebra computationally efficient. Learn what they are all about here, in an intuitive geometric setting, before we discuss (in the next video) how to actually compute a matrix's eigenstuff. This is the sixteenth in what will eventually be a full 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) 0:00 Intro 0:26 Diagonalizing a linear map 5:05 Why diagonal matrices are nice 17:21 Powers of matrices? Who needs 'em? 21:05 Heavenly maps, earthly matrices (Diagonalizing matrices) 23:32 Similar matrices 26:59 Why A = CBC⁻¹ implies that A and B represent the same map. 34:10 Eigendecomposition: Intro 36:04 Interrupting myself with an online matrix calculator 36:18 Getting to know an eigendecomposition 42:13 Symbolic version of eigendecomposition (A=VΛV⁻¹) 44:14 Powers of matrices: Easy with eigendecomposition. 48:28 Summary of eigendecomposition 49:38 One last time: Why does it work? 55:56 Closing thoughts