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I have the following problems when solving a linear equation.

Let $A=(a_{i,j})_{n \times n}$ be a non-negative matrix with $a_{i,j} \in (0,1)$, and let $0 be a scalar. Now we define a vector $x=(x_i)$ of length $n$ as follows:

(I) The first component of $x$ is 1, that is $x_1=1$.

(II) The other components of $x$ (except the first entry of $x$) satisfies the following equation:

$$r \cdot Ax=x .$$

Or equivalently, both (I) and (II) tell that $x$ satisfies the following equation: $$max \{r \cdot Ax,e_1\} =x $$ where $max$ is entry-wise maximum operator, and $e_1={(1,0,\cdots,0)}^T$ .

Based on such a defintion of $x$, I want study the relations between $x$ and the vector $y$ that satisfies $r \cdot Ay=y$ (including the first entry of $y$). In other words, can we compute $x$ from $y$ ?

I would really appreciate any suggestions.

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    I've retagged as it was in no way related to Combinatorics or Numerical Methods.. For the question: What exactly is the dimension of $x = (x_i)$ when $i \neq 1$? Note that it shouldn't be $(n-1) \times 1$ as you're multiplying it with an $n \times n$ matrix.2012-09-01
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    I think you should add "numeric" tag since it is hard to solve $x$ algebrically.2012-09-01
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    I still don't understand the question.. Can you give some example describing what's happening here?2012-09-01
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    John: Where are we at? If you consider that this question is a lost cause, then erase it. Otherwise, modify it, addressing the concerns raised in the comments.2012-09-09

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