# Power iteration for generalized eigenvalue problem $Ax = \lambda B x$ where $A$ and $B$ are singular

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 $$(A, B)$$?

• 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. – lisyarus Aug 30 '19 at 9:50
• @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. – aam Aug 30 '19 at 15:32
• Those look symmetric to me! – lisyarus Aug 30 '19 at 15:36
• Before applying the numerical algorithm, what are the eigenpairs for your system $(A, B)$ (where $B$ is of rank 1)? – user7440 Aug 30 '19 at 21:08
• 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. – user7440 Aug 30 '19 at 21:10