What is an eigenvector?

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  repeatedly applied to ℝ².

In the above,   and . (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 then  doubles in length and points backwards. If λᵢ = 1 then |eig| stays the same. And so on. Above, λ₁ = 1 since  stayed the same length.

It’s nice to add that  and .

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.

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?

• helicopter stability
• quantum particles (the Von Neumann formalism)
• guided missiles
• PageRank 1 2
• the fibonacci sequence
• eigenfaces
• 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.