Eigenvalues and Jordan canonical form of a successively rank-one updated complex matrix with applications to Google's PageRank problem

Let A   be an n×nn×n complex matrix with eigenvalues λ1,…,λnλ1,…,λn counting algebraic multiplicities. Let X=[x1,…,xk]X=[x1,…,xk] be a rank-k   matrix such that x1,…,xkx1,…,xk are right eigenvectors of A   corresponding to λ1,…,λkλ1,…,λk for 1⩽k⩽n1⩽k⩽n, respectively, and V=[v1,…,vk]∈Cn×kV=[v1,…,vk]∈Cn×k be complex matrix. The eigenvalues and Jordan canonical form of the complex matrix A+∑i=1kxiviH are derived. The applications of our results to Google's PageRank problem are also discussed.