This is my homework. I was asked to find all eigenvectors of a symmetric and positive definite matrix by inverse power method with shifted. I encountered three problems:
The eigenvalues to the matrix may not be distinct. In this case, how to find all eigenvectors corresponding to one eigenvalue?
By the inverse power method, I can find the smallest eigenvalue and eigenvector. However, it seems the inverse power method with deflation does not work for finding other eigenvalues. I think the reason is the deflation shift the largest eigenvalue to zero, and after I inverse the matrix, this eigenvalue goes to infinity and therefore it is impossible to find an infinite large eigenvalue. How to modified power method with deflation in the inverse case?
How to use power method with deflation to find all eigenvector? I can find the largest (power method) and second largest (power method with deflation). My question is how to construct the "modifed matrix", i.e. $B = A$ minus some modifying term?