How do we interpret a data matrix, $X$ which does not have $1s$ as the first column? Does it refer to No Intercept Form?

Could it also be interpreted as the mean deviated form? I understand that for mean deviation form, $({X'X})^{-1}$ matrix has order $k-1*k-1$, but I cannot seem to derive the mean deviated form of $(X'y)$ which has order $k-1*1$. Please help.