Eigenvalues and Eigenvectors, Explained Without the Textbook Notation
An eigenvector is a direction a transformation doesn't rotate — it only stretches or shrinks it. The eigenvalue is just the amount of that stretch.
Picture stretching a rubber sheet — pulling it wider left-to-right while leaving the top-to-bottom direction untouched. Almost every arrow drawn on that sheet ends up pointing in some new, rotated direction after the stretch. But two directions don't rotate at all: an arrow pointing straight left-to-right just gets longer, and an arrow pointing straight top-to-bottom stays exactly the same length and direction. Those two special, non-rotating directions are the eigenvectors of that stretch. The number describing how much each one stretched — say, doubled in length, or unchanged — is its eigenvalue.
That's the entire concept: for a given transformation (a stretch, a squeeze, a more complicated combination of the two), an eigenvector is a direction that only gets longer or shorter, never rotated, and the eigenvalue is exactly how much longer or shorter it got. "Eigen" is German for "own" or "characteristic" — these are the directions and stretch amounts that are characteristic of that specific transformation, its own natural axes, in a sense.
Every other direction on the sheet does get rotated by the same transformation, which is precisely what makes the eigenvectors worth naming: they're the special, simple exceptions to what's otherwise a more complicated, harder-to-describe rotation-plus-stretch. If you know a transformation's eigenvectors and eigenvalues, you know its simplest possible description — everything else about how it warps space can be worked out from those two directions and two numbers.
This matters to programmers in places that don't look like linear algebra on the surface. Search engines and recommendation systems use eigenvectors to find the "most stable" pattern in a huge grid of relationships — Google's original PageRank algorithm, at its mathematical core, is finding the eigenvector of a giant matrix representing which web pages link to which. Data-compression and image-processing tools use eigenvectors to find the handful of directions that capture almost all the meaningful variation in a dataset, discarding the rest without losing much — a technique called principal component analysis, which is eigenvectors applied to real-world data.
The notation in a textbook — Av = λv, a transformation matrix A times an eigenvector v equals a number λ times that same vector v — is just this same idea written in symbols: apply the transformation to this special direction, and you get back the same direction, only scaled by λ. Once the rubber-sheet picture is in place, that equation is a restatement of something already understood, not a new concept to memorize on its own.