0
$\begingroup$

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.

Improve this question
$\endgroup$

3 Answers 3

Reset to default
0
$\begingroup$

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.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Thank you. This is precisely what I am looking for. Do you know if matlab or R have any implementations of this ? $\endgroup$
    – user12463
    Commented Jan 24, 2011 at 22:22
  • $\begingroup$ I much doubt that this is implemented in any "standard" packages. $\endgroup$
    – Michael Renardy
    Commented Jan 25, 2011 at 0:14
2
$\begingroup$

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...

Improve this answer
$\endgroup$
1
$\begingroup$

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 $\lambda$.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .