A linear transformation usually changes both the length and direction of a vector. Some special directions behave much more simply: the transformation only scales them.

A nonzero vector $\mathbf v$ is an eigenvector of a linear transformation $T$ when

$$T(\mathbf v)=\lambda\mathbf v$$

for some scalar $\lambda$. The scalar $\lambda$ is the corresponding eigenvalue.

What the eigenvalue means

If $\lambda>1$, the eigenvector is stretched. If $0<\lambda<1$, it is shortened. A negative eigenvalue also reverses its direction.

If

$$\lambda=1,$$

the vector is unchanged. If

$$\lambda=0,$$

the eigenvector is sent to the zero vector.

Eigenvectors are directions, not isolated vectors

If $\mathbf v$ is an eigenvector with eigenvalue $\lambda$, then every nonzero scalar multiple $c\mathbf v$ is also an eigenvector with the same eigenvalue:

$$T(c\mathbf v)=cT(\mathbf v)=\lambda(c\mathbf v).$$

So an eigenvector represents an invariant direction rather than one particular arrow.

Eigenspaces

For a fixed eigenvalue $\lambda$, the vectors satisfying

$$T(\mathbf v)=\lambda\mathbf v$$

together with the zero vector form the eigenspace associated with $\lambda$.

Eigenvectors reveal directions in which a complicated transformation reduces to simple scalar multiplication. Finding enough of these directions can make the whole transformation much easier to understand.

Visual intuition: eigenvectors and eigenvalues