Using original centroid as cluster identifier after applying PCA

Question

Take a look at my original data. (masked with purely random alphabetic here) :

    a b c d e
    f g h i j
A = k l m n o
    p q r s t
    u v w x y

I'm running kMeans (3 cluster) on the data, resulting final centroid like this :

    aa aa aa aa aa
B = bb bb bb bb bb
    cc cc cc cc cc

Now, before run kMean again, I applied PCA on data and took only first three principal component :

    xa yb zc
    xd ye zf
C = xg yh zi
    xj yk zl
    xm yn zo

After that, I ran kMeans for 3 centroid and, of course, resulting 3 centroid :

    xaa yaa zcc
D = xbb ybb zbb
    xcc ycc zcc

The cluster result with PCA are exactly same with the first test (without PCA). My question : After finishing kMeans with PCA, can I say that this cluster (say cluster 1) has centroid aa aa aa aa aa, rather than saying that cluster 1 has centroid xaa yaa zcc?

Answer 1

By definition of a linear transformation (and PCA yields a linear transformation),

$$ M\cdot \mu = M\cdot (\frac{1}{N}\sum_i \vec{x}_i) = \frac{1}{N}\sum_i M\cdot\vec{x}_i $$

I.e. if objects are assigned to the same clusters, then the projection of the original centroid must be the same as the centroid of the projections.

That means PCA did not change anything. Did it work?