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Generalized eigenvalue problem can be reduced to eigenvalue problem $B^{-1} A x = \lambda x$ if $B$ is non-singular matrix. Then, the standard power iteration method can be applied.

How can I use the power iteration method to find the largest eigenvalue and eigenvector of pencil $(A, B)$ if $A$ and $B$ are singular? If it is not possible to use the power iteration method, which computationally efficient method can I use to find the largest eigenvector of pencil $(A, B)$?

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    $\begingroup$ Just to note: for symmetric (or self-adjoint) $A$ and $B$ the $B^{-1}A$ method is bad, since $B^{-1}A$ wouldn't in general be self-adjoint. This is usually solved using Cholesky decomposition $B=LL^T$ and solving $L^{-1}AL^{-T}x=\lambda x$, benefiting from $L^{-1}AL^{-T}$ being self-adjoint. $\endgroup$ – lisyarus Aug 30 '19 at 9:50
  • $\begingroup$ @lisyarus actually, for my problem $A=\sum_i^k u_k u_k^T$ is rank $k < n$ and $B = v v^T$ is a rank-one matrix. $\endgroup$ – aam Aug 30 '19 at 15:32
  • $\begingroup$ Those look symmetric to me! $\endgroup$ – lisyarus Aug 30 '19 at 15:36
  • $\begingroup$ Before applying the numerical algorithm, what are the eigenpairs for your system $(A, B)$ (where $B$ is of rank 1)? $\endgroup$ – user7440 Aug 30 '19 at 21:08
  • $\begingroup$ When B is symmetric positive definite, the power iteration is 1. pick $x_0$; 2 $y_k = B^{-1} A x_k$; 3. $x_{k+1} = y_k / \sqrt{ y_k^T B y_k }$; 4. go to Step 2 - if needed. $\endgroup$ – user7440 Aug 30 '19 at 21:10

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