The eigenvectors of a matrix **summarise** what it does.

- 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.

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

then |**λ**ᵢ = 1.3**eig****ᵢ**| grows by**30%**. If**λᵢ = −2**then doubles in length and points backwards. If**λᵢ = 1**then |**eig****ᵢ**| 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?

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.