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Solving linear system when 1one eigenvalue is known

I have a huge sparse linear system Ax = b$Ax = b$ where I know that an eigenvalue/eigenvector pair is 1$1$ and a vector of all 1's$1$'s. Can this knowledge help me in solving the linear system at all? It seems like it should.

Solving linear system when 1 eigenvalue is known

I have a huge sparse linear system Ax = b where I know that an eigenvalue/eigenvector pair is 1 and a vector of all 1's. Can this knowledge help me in solving the linear system at all? It seems like it should.

Solving linear system when one eigenvalue is known

I have a huge sparse linear system $Ax = b$ where I know that an eigenvalue/eigenvector pair is $1$ and a vector of all $1$'s. Can this knowledge help me in solving the linear system at all? It seems like it should.

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Solving linear system when 1 eigenvalue is known

I have a huge sparse linear system Ax = b where I know that an eigenvalue/eigenvector pair is 1 and a vector of all 1's. Can this knowledge help me in solving the linear system at all? It seems like it should.