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The eigenvectors of a matrix summarise what it does.

  1. Think about a large, not-sparse matrix. A lot of computations are implied in that block of numbers. Some of those computations might overlap each other—2 steps forward, 1 step back, 3 steps left, 4 steps right … that kind of thing, but in 400 dimensions. The eigenvectors aim at the end result of it all.
     
  2. The eigenvectors point in the same direction before & after a linear transformation is applied. (& they are the only vectors that do so) 

    For example, consider a shear three-elevenths shear to the east, per northward block repeatedly applied to ℝ².

    image
    In the above, eig_1 = \vec{blue} = \vec{(1  0)}  and image. (The red arrow is not an eigenvector because it shifted over.)

  3. The eigenvalues say how their eigenvectors scale during the transformation, and if they turn around.

    If λᵢ = 1.3 then |eig| grows by 30%.
     If λᵢ = −2»_i = 2 then eig_i doubles in length and points backwards. If λᵢ = 1 then |eig| stays the same. And so on. Above, λ₁ = 1 since eig_1 = \vec{blue} = \vec{(1  0)} stayed the same length.

    It’s nice to add that image and image.

For a long time I wrongly thought an eigenvector was, like, its own thing. But it’s not. Eigenvectors are a way of talking about a (linear) transform / operator. So eigenvectors are always the eigenvectors of some transform. Not their own thing.

Put another way: eigenvectors and eigenvalues are a short, universally comparable way of summarising a square matrix. Looking at just the eigenvalues (the spectrum) tells you more relevant detail about the matrix, faster, than trying to understand the entire block-of-numbers and how the parts of the block interrelate. Looking at the eigenvectors tells you where repeated applications of the transform will “leak” (if they leak at all).

To recap: eigenvectors are unaffected by the matrix transform; they simplify the matrix transform; and the λ's tell you how much the |eig|’s change under the transform.

Now a payoff.

Dynamical Systems make sense now.

If repeated applications of a matrix = a dynamical system, then the eigenvalues explain the system’s long-term behaviour.

image

I.e., they tell you whether and how the system stabilises, or … doesn’t stabilise.

Dynamical systems model interrelated systems like ecosystems, human relationships, or weather. They also unravel mutual causation.

What else can I do with eigenvectors?

Eigenvectors can help you understand:

  • helicopter stability
  • quantum particles (the Von Neumann formalism)
  • guided missiles
  • PageRank 1 2
  • the fibonacci sequence
  • your Facebook friend network
  • eigenfaces
  • lots of academic crap
  • graph theory
  • mathematical models of love
  • electrical circuits
  • JPEG compression 1 2
  • markov processes
  • operators & spectra
  • weather
  • fluid dynamics
  • systems of ODE’s … well, they’re just continuous-time dynamical systems
  • principal components analysis in statistics
  • for example principal components (eigenvalues after varimax rotation of the correlation matrix) were used to try to identify the dimensions of brand personality

Plus, maybe you will have a cool idea or see something in your life differently if you understand eigenvectors intuitively.

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  14. lemy reblogged this from proofmathisbeautiful and added:
    mind = blown. so that’s why these things are important (outside of rootfinding problems).
  15. meowmeowmeowmeow reblogged this from proofmathisbeautiful
  16. horsestache reblogged this from proofmathisbeautiful and added:
    I’m meant to know this :/
  17. ellyjellybean reblogged this from mathmajorsloth and added:
    I…don’t really understand this, but I’d like to learn :D (Wikipedia, your explanation is quite complex)
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  19. isomorphismes posted this