Let $e_m$ be standard basis vectors for $\mathbb{R}^n$, let $A\in \mathbb{R}^{n \times n}$ be positive definite matrix, and $v\in \mathbb{R}^n$.
Now let vector $u$ be defined as \begin{align} u_m=(e_m^T A v)^k, \, m=1,\ldots, n \end{align} where $k$ is some positive integer.
Question: Can the above expression be re-written in some other equivalent form? Specifically, can we re-write the about in terms $u$ and not individual coordinates of $u$.