Suppose I have a matrix $ Y \in \mathbb{R}^{N \times T}$ where each column $y_t \in \mathbb{R}^{N}$ is a measurement of $N$ different quantities. Assuming the matrix $Y$ is zero-mean on each row, we can compute its covariance matrix $C_y$ as $C_y = \frac{1}{T}YY^\top$. If I want to generate new data $\hat{Y}$ coming from the same distribution (at least same mean and covariance) of the available $Y$ I can simply perform the following operation:
$$\hat{Y} = AX$$
where $A$ is any matrix of appropriate dimensions such that $AA^\top = C_y$ and $X \sim {\cal{N}}(0, I)$ is white Gaussian noise.
Question 1) [SOLVED] If I compute the covariance matrix of $\hat{Y}$ I have: $$C_{\hat{y}}= \frac{1}{T}\hat{Y}\hat{Y}^\top = \frac{1}{T}AXX^\top A^\top = \frac{1}{T}A A^\top = \frac{1}{T} C_y $$ Is that correct? It seems by simulating them on the laptop that my $C_{\hat{y}} = C_y$ and it is not a scaled version of $C_y$. Where am I wrong?
EDIT: I found the mistake here. I was wrongly concluding $XX^\top=I$ because we usually write $\mathbb{E}[XX^\top] = I$ while logically it is $\frac{1}{T}XX^\top=I$.That is:
$$C_{\hat{y}}= \frac{1}{T}\hat{Y}\hat{Y}^\top = A \frac{1}{T} XX^\top A^\top = A A^\top = C_y $$
Question 2) Suppose I want to fit a matrix $H$ to the generated data as $\hat{Y} = H X$, where the $X$ is the same matrix used to generate the $\hat{Y}$ in the previous step. My intuition is that here I do not suffer from the unitary freedom "problem". In other words the problem:
$$\min_H \|\hat{Y} - HX\|_F = \min_H \|(A - H)X\|_F = \min_H \|A - H\|_F $$ is in my view well-defined, and I do not need in the problem another (unitary) matrix $Q$ as in: $$ \min_H \|A - HQ\|_F \quad \text{ s.t. } QQ^\top = I.$$
Clearly such matrix $Q$ enable the mapping $HQ$ to be more expressive than the simple $H$ alone, but it should not be needed.
I was wondering this because if I try to fit the covariance matrix directly as $C_\hat{y} = HH^\top$ then I suffer of the unitary ambiguity $\|A - HQ\|_F$, but I do not see the link with my problem above. Could you please correct and comment my reasoning ? Thanks