Eigenvectors and eigenvalues

Explained Visually

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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.

012345012345xyva₁a₂Αv

a₁,x

1.00

a₂,x

0.50

a₁,y

0.50

a₂,y

1.00

0.50

1.00

2.00

v, x

3.00

v, y

Αv

3.50

v, x

4.00

v, y

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.

Av=(1821)⋅(12)=5(12)=λv.

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.

012345012345xyva₁a₂Αvλ₁ = 1.5λ₂ = 0.5s₁s₂

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.

02004006008001,00002004006008001,000xyvΑvΑ²va₁a₂λ₁ = 1.1λ₂ = 0.5s₁s₂

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

adultst+1childrent+1==adultst+childrentadultst

which we can rewrite in matrix form like

vt+1(adultst+1childrent+1)==A(1110)⋅⋅vt(adultstchildrent)

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