Example 9. For the set $\{1,2,3, \cdots, n\}$ and any of its non-empty subsets, define the "alternating sum" as follows: rearrange the elements in decreasing order, then alternately add and subtract the subsequent numbers starting from the largest. For example, for $\{1,2,4,6,9\}$, rearranged as $(9,6,4,2,1)$, the alternating sum is $9-6+4-2+1=6$. The alternating sum of $\{5\}$ is 5. For $n=7$, find the total of all alternating sums.
Solution: Let's define the alternating sum of the elements of the empty set to be 0. Divide the $2^{n}$ subsets of $\{1,2, \cdots, n\}$ into two categories: one category contains $n$ and has $2^{n-1}$ subsets, and the other category does not contain $n$, also having $2^{n-1}$ subsets. Pair the subset $F=\left\{n, a_{1}, a_{2}, \cdots, a_{i}\right\}$ containing $n$ with the subset $F^{\prime}=\left\{a_{1}, a_{2}, \cdots, a_{i}\right\}$ not containing $n$. Clearly, this pairing is one-to-one, and the sum of the alternating sums of the two subsets in each pair is $n$. Therefore, the total alternating sum $S$ of all elements of $\{1,2, \cdots, n\}$ is $n \cdot 2^{n-1}$. Thus, when $n=7$, the total alternating sum is 448.