Are these two statements equivalent:
$ \sum\limits_{i=1}^{n} \alpha = \sum\limits_{i=1}^n \sum\limits_{j=1}^m A_{i,j}b_{i} $
and
$ \alpha = \sum\limits_{j=1}^m A_{i,j}b_{i} $
where $\alpha$ is a constant, $b$ is a vector and $A$ is a matrix. Something I'm reading uses this equivalency in their derivations but the reason why this doesn't sit well with me is because the first seems like a more relaxed condition (it's restricting summation over both rows and columns), whereas second is stricter (it's restricting summation over each column).
I realize this is probably too simple a question but would appreciate any help.
Edit: I should add that we know $\sum\limits_{i}^n b_i = 1$ and $\sum\limits_{i}^n\sum\limits_{j}^m A_{i,j} = 1$
In a general sense, no, the two statements are not equivalent. However, there are specific scenarions where the equivalence could hold true.
So let's first answer the question of whether they are always equivalent:
The first statement:
$$\sum_{i=1}^n\alpha = \sum_{i=1}^n\sum_{j=1}^mA_{i,j}b_i$$
sums the constant $\alpha$ over $i$ from 1 to $n$, effectively multiplying $\alpha$ by $n$. It implies that the sum of $\alpha$ over $n$ terms equals the sum of products of $A$'s entries with $b$'s entries over all rows and columns, as you pointed out. It's a statement about the entire matrix $A$ and the vector $b$.
The second statement:
$$\alpha = \sum_{j=1}^m A_{i, j}b_i$$
equates $\alpha$ to the sum over $j$ of the product of the matrix $A$'s entries and the vector $b$'s entries for a fixed $i$. It is a per-row statement, saying that for each row $i$, $\alpha$ equals the sum of the products of the entries of that row with the corresponding entries of $b$.
Therefore, as you correctly pointed out, they describe different relationships, and are not, at face value, equivalent.
Let us now look at some scenarios where specific conditions of $A$, $b$, or $\alpha$ could make them equivalent:
If each row of the matrix $A$ multiplies with vector $b$ to give the same sum, which is $\alpha$, the equivalence holds.
If the matrix $A$ and vector $b$ are structured in such a way that the product of each row of $A$ with $b$ yields the same scalar $\alpha$, the equivalence holds.
If $\alpha$ is defined or derived in a way that makes it equal to the sum of products for each row of $A$ and $b$, the equivalence holds.
Hope this helps. Good luck with your studies!