Matrix representation of PCA

Question

In both Wikipedia and this medium post, I see the succinct principle components decomposition of X represented as

$$T=XW$$

However, it seems to me that it should be $T = WX$ instead, if according to the Wikipedia page that columns of $W$ are the eigenvectors of $X^TX$ and $X$ is arranged as row vectors for each observation. In short, $X \in \mathbb{R}^{nxk}$, where $n$ is the number of observations and $k$ is the dimension of each data point, and $W \in \mathbb{R}^{nxn}$, so $XW$ does not even exist.

Am I getting the derivation somewhere wrong?

Answer 1

Since the data samples have dimension $k$, the scatter matrix, $X^TX$ will be of dimension $k\times k$ leading to $k$ dimensional eigenvectors. So, $W$ is of dimension $k\times k$. With this correction, $T=XW$ indeed exists. You can also think of it as $$T=\underbrace{\begin{bmatrix}x_1^T\\x_2^T\\\vdots\\x_n^T\end{bmatrix}}_{X^T}\underbrace{\begin{bmatrix}w_1&\cdots&w_k\end{bmatrix}}_{W}=\begin{bmatrix}x_1^Tw_1&\cdots &x_1^Tw_k\\\vdots&\ddots&\vdots\\x_n^Tw_1&\cdots&x_n^Tw_k\end{bmatrix}$$ where $i$-th row of $T$ represents the projection of $i$-th data sample to the first $k$ (whatever it is) PCs.