I'm fairly new to proofs with matrices, so determining when I'm allowed to use certain indices to denote arbitrary entries in various matrices is still a challenge. For this problem, I want to prove that the column-row expansion of two matrices $A$ and $B$ is equal to the product matrix $AB$.
Let $A$ be an $m \times r$ and $B$ an $r \times n$ matrices. Partition $A$ into $r$ column vectors $\mathbf{c}_k$ and partition $B$ into $r$ row vectors $\mathbf{r}_k$.
The product $AB$ is an $m\times n$ matrix where each term is defined by
$$(AB)_{ij} = \sum_{k=1}^r a_{ik}b_{kj}$$
Now consider the sum $\sum_{k=1}^r \mathbf{c}_k \mathbf{r}_k$of each column-row of $A$ and $B$, respectively. Since each $\mathbf{c}_k\mathbf{r}_k$ has size $m\times n$, it follows that the sum also has dimension $m\times n$.
Now the entries of the sum have the form
\begin{align}\big(\sum_{k=1}^r\mathbf{c}_k\mathbf{r}_k \big)_{ij} &= \sum_{k=1}^r(\mathbf{c}_k\mathbf{r}_k)_{ij} \\ &= \sum_{k=1}^r (a_k b_k)_{ij} \\ &= \sum_{k=1}^r(a_{ik}b_{kj})\end{align}
So the two expressions are equal.
My main concern is with the use of indices in the second equation, specifically the last equation $\sum_{k=1}^r(a_kb_k)_{ij} = \sum_{k=1}^r(a_{ik} b_{kj})$.
If there is nothing wrong, help me see how this step is justified. Otherwise, please let me know what I can do to fix it. Thank you.
Best Answer
Your proof looks correct, except that you should also mention that $a_k$ is the $k$th column of $A$ and $b_k$ is the $k$th row of $B$.