In the video that you have, you can see, graphically, an example of the eigenvalues and the autovectors of functions. Graphically explains very well what is an eigenvector (you know that using the matrix of linear application gives us the same by a value), ie:
Where A is the matrix of the linear application and is a vector of the source space that, as you see, when transformed, gives the same multiplied by a value (which can be a real or complex number).
As you can see, although eigenvalues and autovectors are mainly used to diagonalize a matrix, to know if it is diagonalizable and to remove the diagonalization spaces, they also help us, with the linear application, to transform any vector by taking the autovectors as a basis for not transforming and using The eigenvalues to calculate the vector transformed by the application. Obviously, it is the logical conclusion of a diagonalization and space (you know, the base) that is not transformed under diagonalization.
How beautiful is algebra!