Eigenstuff

Imagine you have a small collection of things {…}. You take linear combinations (in the spirit of “When Doves Cry inside a Convex Hull”) of them, making varied and interesting combinations that explore — even span — a “space”.

Maybe it’s 

  • the space of all possible faces
  • the space of all possible songs
  • the space of all possible configurations of a heat engine
  • the space of all possible orientations of a sculpture
  • the space of all possible paintings
  • the space of all possible artistic schools
  • the space of all possible states of your desk
  • the space of all possible functions ℝ⁴→ℝ
  • the space of all possible functions ℝ[0,1]→ℝ[0,1]
  • the space of all possible directions in 3-D
  • a subset of the space of all possible graphs
  • the space of all possible quantum configurations of a molecule
  • the space of all possible notes
  • the space of all possible ways a week can go

(It’s possible to mathematicise these things in part because of the modern, abstract notion of a “vector”.)

Think about this large space instead of the original small-collection-of-things {…} that generated it. Could a different small-collection-of-things have spanned the exact same space? Surely so.

What about different small-collections {..A..}, {..B..} that could have generated the space? Is there a really simple collection? One whose parts don’t self-interfere. One that’s representative and easy to work with.

If there is such a “canonical basis”, the elements of the basis are called eigenfaces, eigenmodes, eigengraphs, eigenstates, or some other kind of eigenbasis.

hydrogen electron states S, P, D, F which are eigenstates

Eigenboogers are fundamental to how abstract mathematics made “linear” algebra hugely relevant — to ODE’s, image compression, spectral analysis, crystallography, economics, statistical analysis, geophysics, and teaching artificial intelligence how to spot terrorists at the Superbowl.

Finally, eigenstates and eigenmodes are pretty good for meditating on how the universe is a song, singing itself.

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