Eigenvectors and Eigenvalues are those vectors don’t change direction only stretching with the multiplication of eigenvalue.
For a transformation that have eigenvectors can span the space, we can use eigenvectors as basis.
If we do so, we can apply the eigenvectors to this transformation. The result must be a diagonal matrix, and the value equal to the eigenvalues. (eigenvector only stretch on its direction, so the value only shows on the diagonal)
If we want to calculate 100-th power of this matrix, we can convert it to its system and calculate, then convert it back.
set of eigenvectors is also called eigenbasis.