# How can finite products of finite sums, $\prod_{k=0}^M \left \{ \sum_{n=0}^N a_{n,k} \right \}$, be represented in matrix form?

If I have an expression such as :

$$\prod_{k=0}^K \left \{ \sum_{n=0}^N a_{n,k} \right \}$$

If we expand a bit, we get:

$$\prod_{k=0}^M \left \{ a_{0,k} +a_{1,k}+...+a_{N-1,k}+ a_{N,k} \right \}$$ $$(a_{0,0} +a_{1,0}+...+a_{N-1,0}+ a_{N,0})(a_{0,1} +a_{1,1}+...+a_{N-1,1}+ a_{N,1})...(a_{0,M} +a_{1,M}+...+a_{N-1,M}+ a_{N,M})$$ How can this be represented concisely in matrix form? Any other interesting properties/theorems on this (i.e. convergence for when M and/or N $$\rightarrow \infty$$...etc.)?

Let $$u=(1,1,...,1)^T$$ be the $$N$$-dimensional vector with $$1$$ as its components, and $$a_k=(a_{0,k}, a_{1,k},...,a_{N,k})^T$$.
If $$K=0$$, then the product is $$\langle a_0, u\rangle=a_0^Tu=u^Ta_0$$.
If $$K=1$$, then the product is $$a_0^Tuu^Ta_1=a_0^TMa_1$$, where $$M=u^Tu=u\otimes u$$ is the $$N\times N$$ matrix full of $$1$$s.
For $$K>1$$, then the product cannot be simply expressed with matrices (unless you want to keep the product somewhere), but you can use the tensor $$M=u\otimes u\otimes \ ... \, \otimes \, u$$ with $$1$$s as entries.