Eigenvalues and Jordan canonical form of a successively rank-one updated complex matrix with applications to Google's PageRank problem (Q929938)

Let \(A\) be an \(n\times n\) complex matrix with eigenvalues \(\lambda_1,\dots,\lambda_n\) counting algebraic multiplicities. Let \(X=[x_1,\dots,x_k]\) be a rank-\(k\) matrix such that \(x_1,\dots,x_k\) are right eigenvectors of \(A\) corresponding to \(\lambda_1,\dots,\lambda_k\) for \(1\leq k\leq n\), respectively, and \(V=[v_1,\dots,v_k]\in \mathbf{C}^{n\times k}\) be a complex matrix. The author derives the eigenvalues and the Jordan canonical form of the complex matrix \(A+\sum_{i=1}^k x_iv_i^H\), and the latter result is used to describe the Jordan canonical form of the generalized Google matrix \(\tilde{A}(c,c_1,\dots,c_k)=cA+\sum_{i=1}^k(c_i\lambda_i)x_iv_i^H\), where \(c,c_i\in\mathbf{C}\), \(1\leq i\leq k\), and \(c+\sum_{i=1}^kc_i=1\).