# Is there a reason to do PCA on disjoint subsets of variables rather than all the variables?

Principle component analysis (PCA) is a coordinate transformation of the original data, so that the projection of the data onto the first new coordinate has the greatest variance. I want to know whether there are situations where it is advantageous to split the variables into disjoint subsets, and doing PCA on each subset, instead of doing one PCA on all the data.

In matrix notation, suppose we have a data matrix $M$, of size $n_{\text{obs}} \times p$. That is, the number of rows in $M$ is equal to the number of observations, and the number of columns equal to the number of variables. The new coordinates for each observation is given by $MT$, where $T$ is a $p \times p$ matrix. If the variables are split into subsets, and arranged appropriately, $T$ will be block diagonal.