A linear transformation T that stays in the same vector space could have some points that stay relatively fixed. A fixed vector v would have T(v)=v. Many transformations only have the 0 vector as their only fixed point.
But, what happens in a lot is that there are certain vectors that get scaled by the transformation. That is, there exists a scalar λ such that T(v)= λv. Since all scalars are unaffected by T, this same λ works for all scalings of v. If v has this property, it is an eigenvector for T, and λ is an eigenvalue.
A transformation can have multiple independent eigenvectors.
Another cool property is that if λ is less than 1, then T repels vectors from v, and if λ is greater than 1, then it attracts T.
This is an equal superposition of the |3,2,2> and |3,1,-1> eigenstates:
This is an equal superposition of the |4,3,3> and |4,1,0> eigenstates:
What is an eigenstate? It’s a convenient state to use as a basis. We get to decide which quantum states are “pure” and which “mixed”. There’s an easy way and a hard way; the easy way is to use eigenstates as the pure states.
More mathematically: the Schrödinger equation tells us what’s going on in an atom. The answers to the Schrödinger equation are complex and hard to compare. But phrasing the answers as combinations of eigenfunctions makes them comparable. One atom is 30% state A, 15% state B, 22% state C … and another atom is different percentages of the same states.
Eigenvalues/vectors are instrumental to understanding electrical circuits, mechanical systems, ecology and even Google's PageRankalgorithm. Let’s see if visualization can make these ideas more intuitive