Expected Value of Product of Matrices

Question

Suppose we have a data matrix $X = [X_{ij}]$ belongs to $\mathbb{R}^{d \times N}$ i.e., the number of data samples is $N$ and each column corresponds to $d$-dimensional single data in $\mathbb{R}$. Under the Principle Component Analysis (PCA), the data matrix is mapped into a smaller matrix $Y$ belongs to $\mathbb{R}^{k \times N}$ where $k$ is less than $d$ as $Y = AX$ for some matrix $A$. Show that if the expectation of each dimension of the original data $X$ is zero, then that of $Y$ is also zero for any matrix $A$.

Answer 1

Just like BGM said.

$Y = AX \implies y_{ij} = \sum\limits_{k}{a_{ik}x_{kj}}$

The expectation of elements in $X$ is zero: $\mathbb{E} x_{kj} = 0$   $\forall k, j$, hence

$\mathbb{E} y_{ij} = \sum\limits_{k}{a_{ik} \mathbb{E} x_{kj}} = 0$.