### Eigenvectors and eigenvalues explained visually

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# Eigenvectors and eigenvalues

### Explained Visually

By Victor Powell and Lewis Lehe

Eigenvalues/vectors are instrumental to understanding electrical circuits, mechanical systems, ecology and even Google's PageRank algorithm. Let's see if visualization can make these ideas more intuitive.

To begin, let v be a 2-dimensional vector (shown as a point) and A be a matrix with columns a1 and a2 (shown as arrows). If we multiply v by A, then A sends v to a new vector Av.

If you can draw a line through (0,0), v and Av, then Av is just v multiplied by a number λ; that is, Av=λv. In this case, we call λ an **eigenvalue** and v an **eigenvector**. For example, here (1,2) is an eigvector and 5 an eigenvalue.

Below, change the bases of A and drag v to be its eigenvector. Note two facts: First, every point on the same line as an eigenvector is another eigenvector. That line is an **eigenspace**. Second, when λ<1, Av is closer to (0,0) than v; and when λ>1, it's farther away.

### What are eigenvalues/vectors good for?

Eigenvalues/vectors explain the behavior of systems that evolve step-by-step, where each step occurs as multiplication by a matrix A. If you keep multiplying v by A, you get a sequence v,Av,A2v, etc. As you can see below, eigenspaces attract this sequence and draw it toward (0,0) or farther away, depending on their eigenvalues.

Let's explore some applications and properties of these sequences.

### Fibonacci Sequence

Suppose you have some amoebas in a petri dish. Every minute, all adult amoebas produce one child amoeba, and all child amoebas grow into adults (Note: this is not really how amoebas reproduce.). So if t is a minute, the equation of this system is

which we can rewrite in matrix form like

Below, press "Forward" to step ahead a minute. The total population is the Fibonacci Sequence.