The context is I have a matrix of feature vectors $x = [x^{(1)}\ \ x^{(2)} \ \cdots \ \ x^{(n)}]^T$ but each $x$ is raised to the powers $0 \leq j \leq M$.
So it looks like this:
$$ \left[ \begin{array}{c} \left(x^T\right)^0 \\ \left(x^T\right)^1 \\ \vdots \\ \left(x^T\right)^M \end{array} \right] $$
But where the powers are element-wise. I guess maybe there is no other way of interpreting the powers.
Actually I think I just realized something, isn't this matrix just a Vandermonde matrix transposed?
Yes, it appears to be a Vandermonde matrix. On Wikipedia they simply used the index notation
$$ V_{i,j} = \alpha_{i}^{j-1} $$