Thanks again for copper.hat and Robert Israel's quick immediate reply. While I am modifying the questions, they've already given the answer. Now in this thread, I've changed it back to the original version, and I put the new version which is more challenging in the following thread: A hard proof of two matrix's elements. Hopefully, some brilliant experts could give me suggestions.
Given an constant $\alpha \in (0,1)$, and an $n \times n$ matrix $X$ whose all entries are between 0 and 1, and ${\|X\|}_{\infty} \le 1$. Suppose $A=\sum_{i=1}^{\infty} {\alpha}^i X^i ,$ $B=\sum_{i=1}^{\infty} \frac {{\alpha}^i}{i!} X^i ,$
I've done some experiments and found that :
- For each entry $(a,b)$, $[A]_{a,b} \ge [B]_{a,b} .$
(Note that I use $[A]_{i,j}$ to denote the $(i,j)$-entry of the matrix $A$)
How can I prove this result mathmetrically? Any suggestions are warmly welcome.