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

Let A be an n × n complex matrix with eigenvalues λ 1 , … , λ n counting algebraic multiplicities. Let X = [ x 1 , … , x k ] be a rank-k matrix such that x 1 , … , x k are right eigenvectors of A corresponding to λ 1 , … , λ k for 1 ⩽ k ⩽ n , respectively, and V = [ v 1 , … , v k ] ∈ C n × k be complex matrix. The eigenvalues and Jordan canonical form of the complex matrix A + ∑ i = 1 k x i v i H are derived. The applications of our results to Googleʹs PageRank problem are also discussed.