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Any reference on how to solve the problem $Ax + c = \lambda Bx$ , where $A$, $B$ are full rank matrices, $c$ and $x$ are vectors and $\lambda$ is an unknown constant. I want to solve for both $x$ and $\lambda$. Without the vector c, this is a Generalized Eigen problem and is easy to solve. I actually don't need the value of $\lambda$ I care only about $x$. I just need a citation and I can follow it up. |
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I think what is missing from the statement of the problem is a normalization condition on x, such as $\|x\|=1$. Problems of this kind are known as inhomogeneous eigenvalue problems, and there is some literature about them. See e.g. R.M.M. Mattheij and G. Soderlind, Linear Algebra and its Applications 88 (1987). p. 507. |
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Since affine transformations (your $Ax + c$) are linear transformations in one dimension higher, this is easy to reduce to the Generalized eigenproblem you know how to solve... |
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Your equation is $(\lambda B-A)x=c$. Since $B$ has full rank, this has a solution for all but finitely many $\lambda$. You can't solve for $x$ and $\lambda$ as there's a solution $x(\lambda)$ for almost all $\lambda$. |
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