I read in a journal recently that it can be easily seen that the Hotelling's one-sample t-test is equivalent to a two-sided one-sample t-test with response variable $\hat{V} = \sum_{i=1}^pa_iv_i, $the projection of a unit vector $\mathbb{1}_n$ into the column space of our data $X$.
Here, I am defining $X$, a $p \times n$ matrix as our data matrix. And that we have:
$$ X = \begin{pmatrix} v_1^T\\ \vdots\\ v_n^T \end{pmatrix} $$
where the rows $v_i = (x_{i1}, x_{i2}, \ldots, x_{in})$ represent the possible response variables $i$.
I am not easily seeing an equivalence here. Does anyone see what is going on? Thanks!