http://setosa.io/ev/eigenvectors-and-eigenvalues/
Explained Visually
By and
Eigenvalues/vectors are instrumental to understanding electrical circuits, mechanical systems, ecology and even Google's 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 .
As you can see, the system goes toward the grey line, which is an eigenspace with λ=(1+5√)/2>1.
Steady States
Suppose that, every year, a fraction p of New Yorkers move to California and a fraction q of Californians move to New York. Drag the circles to decide these fractions and the number starting in each state.
New YorkCalifornia1 − p = 0.7p = 0.3q = 0.11 − q = 0.938.33m19.65mTo understand the system better, we can start by writing it in matrix terms like:
It turns out that a matrix like A, whose rows add up to zero (try it!), is called a , and it always has λ=1 as an eigenvalue. That means there's a value of vt for which Avt=λvt=1vt=vt. At this "steady state," the same number of people move in each direction, and the populations stay the same forever. Hover over the animation to see the system go to the steady state.
Complex eigenvalues
So far we've only looked at systems with real eigenvalues. But looking at the equation Av=λv, who's to say λand v can't have some imaginary part? That it can't be a number? For example,
Here, 1+i is an eigenvalue and (1,i) is an eigenvector.
If a matrix has complex eigenvalues, its sequence spirals around (0,0). To see this, drag A's columns (the arrows) around until you get a spiral. The eigenvalues are plotted in the real/imaginary plane to the right. You'll see that whenever the eigenvalues have an imaginary part, the system spirals, no matter where you start things off.
steps: -3-2-1123-3-2-1123-33-33-33-33realimrealimλ₀λ₁Learning more
We've really only scratched the surface of what linear algebra is all about. To learn more, check out the legendary Gilbert Strang's course at MIT's Open Courseware site. To get more practice with applications of eigenvalues/vectors, also ceck out the excellent course.
For more explanations, visit the Explained Visually
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