Eigenvectors and their meaning 

If all the n eigenvalues of an n-square matrix A are distinct, then the n eigenvectors are linearly independent and the matrix is similar to a diagonal matrix D. This means that matrix A represents the same linear transformation as the diagonal matrix D – it is simply referred to a different basis (i.e. coordinate system). We know the character of the point transformation effected by a diagonal matrix – it represents simply stretching (or compressing) effects directed in the directions of its different coordinate axes. This stretching (or compressing) effected by the diagonal matrix will occur in a coordinate system whose axes correspond to the eigenvectors and with magnitudes given by the eigenvalues .