Consider an $n \times p$ data matrix $\mathbf X$ with mean-centered data (i.e. each variable has mean zero). And suppose that we perform PCA on $\mathbf X$, obtaining principal components of the form: $\mathbf{z = Xv}$, where $\mathbf v$ represents a generic eigenvector of $(1/n)\mathbf X^T \mathbf X$.
Show that the mean of any principal component $\mathbf z$ is zero.
I am very confused on how to prove the principal component of mean centered data is zero. Any explanation would be appreciated.
Hint: A vector $\mathbf w$ will as mean zero if and only if $\mathbf 1^T \mathbf w = 0$, where $\mathbf 1$ is the column-vector $(1,\dots,1)$.