What is the result of the method for multiple eigenvalues? Is there any case for which this method will not work altogether?
I am going to share the answer I've got from my professor.
In the case of multiple eigenvalues, the exact analytical solution for the eigenvector contains a much greater degree of uncertainty than in the case of a non-multiple one.
For a non-multiple eigenvalue, the eigenvector is defined up to its length, while its direction is determined uniquely, and the normalization condition makes the solution unambiguous.
For multiple eigenvalues, there is uncertainty already in the direction of the eigenvector, and the higher the multiplicity, the greater the degree of freedom in the direction (in the limiting case of an n-fold eigenvalue — for example, an n-fold zero for a null matrix-the eigenvector is formally any vector). For a method of inverse iterations with intermediate normalization, there is no single limit vector that the method can converge to.