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.
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.
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.