a minimum and maximum value problem

First and foremost, I greatly appreciated the prior attempts made by the excellent mathematicians Robert Israel, and mixedmath on the related problem. Now I have the following problems of the revised model.

Given a constant $\alpha \in (0,1)$, and an $n \times n$ matrix $X$ whose all entries are between 0 and 1, and each row sum of $X$ is 1, and ${\|X\|}_{\infty} \le 1$.

Suppose $$A=(1- \alpha) \cdot \sum_{i=0}^{\infty} {\alpha}^i X^i ,$$ $$B=\exp(-\alpha) \cdot \sum_{i=0}^{\infty} \frac {{\alpha}^i}{i!} X^i ,$$

I use $[A]_{a,b}$ to denote the $(a,b)$-entry of the matrix $A$.

I want to

(1) find the minimum and maximum values of $${[A-B]}_{a,b} \quad \text{ for all a,b=1,...,n}$$

(2) I've done some experiments and found that :

For every two entries $(a,b)$ and $(c,d)$ , in most cases (but not all cases see here) we have

• if $[A]_{a,b} \ge [A]_{c,d}$, then $[B]_{a,b} \ge [B]_{c,d}.$

Is there any mathmetrical result on my above observation?

Any suggestions for either of the questions are warmly appreciated!

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Isn't it related to Markov Chains somehow? –  Ilya May 31 '12 at 16:03
$X$ is a stochastic matrix. I'm not sure whether Markov chain will help. –  John Smith May 31 '12 at 16:09
I don't know why the OP doesn't mention this, but this is the next question in a series of questions along the same sort of theme. Most recently, math.stackexchange.com/questions/151921/…, math.stackexchange.com/questions/151910/a-matrixs-element-proof, and math.stackexchange.com/questions/144959/…. –  mixedmath May 31 '12 at 16:22
@mixedmath, I aprreciate all the mathsmaticians giving me so many clues and hints to polish my model. As an example, previously I thought $A$ and $B$ have kept the entry order, now your counterexample prove that my initial thoughts being wrong. –  John Smith May 31 '12 at 16:36
If you appreciate what people have done for you, please show your appreciation by linking back to related questions when you post a new one. –  Gerry Myerson Jun 1 '12 at 1:30