Example 4 Given positive numbers $a_{1}, a_{2}, \cdots, a_{n}(n \geqslant 3)$ satisfying
$$
\begin{array}{l}
A>a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n}, \\
a_{1}+a_{2}+\cdots+a_{n}=2 A .
\end{array}
$$

Let $\alpha$ be the minimum of $A-a_{1}, a_{1}-a_{2}, \cdots, a_{n-1}-a_{n}$. Find the maximum value of $\alpha$.