What eigenvectors and eigenvalues mean geometrically

Most vectors change direction under a linear map. An eigenvector is special because $Av=\lambda v$ keeps the vector on the same line through the origin.

If $\lambda >1$, the vector is stretched. If $0<\lambda <1$, it is compressed. If $\lambda <0$, the direction is flipped as well as scaled.

Once you know the invariant directions, repeated applications of the matrix become easier to understand. That is why diagonalization turns matrix powers into powers of eigenvalues.

The geometry and the computation are the same story told two ways: invariant directions on one side, simpler coordinates on the other.

For a generic vector, a linear transformation usually mixes coordinates and changes direction. Eigenvectors are the rare directions where that mixing disappears.

That is why eigenvectors matter geometrically: they identify the directions along which the map behaves like simple scaling instead of a complicated combination of scaling and turning.

A practical test is: pick a vector $v$, compute $Av$, and ask whether $Av$ lies on the same line as $v$. If it does, then $v$ is an eigenvector and the scale factor is the eigenvalue.

If you can collect enough independent eigenvectors, they form a basis adapted to the transformation itself. In that basis the matrix becomes diagonal.

So diagonalization is not a separate topic from eigenvectors. It is the coordinate-system version of the same geometric idea.