Eigenvectors and Eigenvalues

An eigenvector of a matrix is simply a direction the matrix doesn't turn: vectors pointing that way only get stretched, shrunk or flipped. The name is half German. "Eigen" means "own," so an eigenvector is one of the matrix's own special directions, and its eigenvalue is the matrix's own stretch factor along it. They're useful because along those directions a complicated matrix acts like multiplying by a single number. All we're really saying in $A\vec v = \lambda\vec v$ is: $A\vec v$ (what the matrix does to $\vec v$) equals $\lambda\vec v$ (just $\vec v$ scaled by a number $\lambda$), with $\vec v \ne \vec 0$ (the zero vector doesn't count). It's one short equation; bear with it.

Most vectors get knocked off their line by a transformation. Eigenvectors are the special ones that don't. There's an animation further down that shows the difference.

Checking a candidate: compute $A\vec v$ and see whether it's a multiple of $\vec v$.

Eigenspaces. All eigenvectors for a given $\lambda$ (plus $\vec 0$) form a subspace. Think about why. $A\vec v = \lambda\vec v$ is the same as $A\vec v - \lambda\vec v = \vec 0$, or $(A - \lambda I)\vec v = \vec 0$. So the eigenvectors for $\lambda$ are just the null space of $A - \lambda I$, and you already know null spaces are subspaces.

Zero as an eigenvalue means $A\vec v = \vec 0$ for some nonzero $\vec v$: the matrix is not invertible.

A common mistake. It's tempting to think $\vec 0$ is an eigenvector, because $A\vec 0 = \lambda\vec 0$ is always true. It isn't: that equation holds for every $\lambda$ at once, so it tells you nothing, which is why eigenvectors have to be nonzero. The eigenvalue $\lambda = 0$, on the other hand, is perfectly fine. It just means some nonzero vector gets squashed to $\vec 0$.

Google's PageRank is an eigenvector of the web's link matrix; the vibration modes of buildings, bridges and molecules are eigenvectors; principal components of data are eigenvectors.