Example 5 For the set $\{1,2, \cdots, n\}$ and each of its non-empty subsets, we define the "alternating sum" as follows: arrange the numbers in the subset in ascending order, then alternately add and subtract the numbers starting from the largest (for example, the alternating sum of $\{1,2,4,6,9\}$ is $9-6+4-2+1=6$, and the alternating sum of $\{5\}$ is 5). For $n=7$, find the total of all these alternating sums.
The set $\{1,2, \cdots, 7\}$ has $2^{7}-1=127$ non-empty subsets, and each element in $\{1,2, \cdots, 7\}$ appears $2^{6}=64$ times in the subsets. Since $1, 2, \cdots, 6$ appear 32 times in odd positions and 32 times in even positions in the alternating sum, the total sum of these numbers is 0, while 7 appears 64 times and always takes a positive value. Therefore, the total sum of the alternating sums of all subsets is $7 \times 64=448$.