I've generated a dataset of 100 elements from a 3-variate Gaussian distribution with parameters $\mu = 0$ and $\Sigma = \begin{pmatrix}1 & \rho_1 & \rho_2 \\ \rho_1 & 1 & \rho_1 \\ \rho_2 & \rho1 & 1\end{pmatrix}$. To generate such dataset, Cholesky factorization as been used on the covariance matrix $\Sigma$. I followed this for Python language.
The problem is related to adding noise to such dataset. I need to add noise $\eta_i \sim \mathbb{N}(0,1)$ to each $X_i \quad \forall i=1, \dots, 100$ in such a way its covariance matrix results to be $\Sigma* = \begin{pmatrix}1+noise & \rho_1 & \rho_2 \\ \rho_1 & 1+noise & \rho_1 \\ \rho_2 & \rho1 & 1+noise\end{pmatrix}$ that is noise does not alter (or it if does, for a very less amount) all the elements off-diagonal. Is it possible to do it? Or is the sample too small?
I just add the noise in such a way that $X_i + \eta_i$, should I multiply $\eta_i$ for $\mathbb{I}$ first?
Also $\eta$'s size is [100x3]: should it be smaller to have such $\Sigma*$?
Update
My goal is to find such $X^*$, the dataset + noise, such that I can apply to it the whitening and colouring phases (here) and show that the only noise addition cannot preserve privacy of users in the dataset, but a recolouring (whitening + colouring) are mandatory.
Thus, Noise addition is in Chaper 4 of link says that
epsilon is the random variable (noise) added to X, whose distribution is $đ(0,\sigma^2)$. Note that additive noise also known as uncorrelated noise, preserves the mean and covariance of the original data but the correlation coefficients and variances are not sustained. Another variation of additive noise is correlated additive noisethat keeps the mean and allows the sustenance of correlation coefficients in the original data.