Let $\mathbf{a}$ and $\mathbf{u}_i$ $i=1,2,\ldots,n$ be column vectors and define $U$ to be the product $$ U = \prod_{i=1}^{n} \mathbf{a}^\top\mathbf{u}_i. $$ Not sure if this is a trivial question but is there a way to write $U$ as either $$ U = \mathbf{k}^\top \mathbf{x} \qquad \text{or}\qquad U = \mathbf{x}_1^\top \mathbf{A} \mathbf{x}_2 $$ where $\mathbf{k}$ and $\mathbf{A}$ are respectively a column vector and a matrix involving only $\mathbf{a}$ while $\mathbf{x},\ \mathbf{x}_1$ and $\mathbf{x}_2$ are column vectors involving only $\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n$?
I've attempted using the trace operator but I just can't push through.