Independence of distribution

Question

Let there be a random matrix defined as $\mathbf{H}_1 = X + \boldsymbol\nu$, where, $X$ is deterministic and $\boldsymbol\nu$ is Gaussian white noise. Now let there be another random matrix defined as $\mathbf{H}_2 = Y + \boldsymbol\nu$, where $Y$ is deterministic. Here $X$ and $Y$ are different matrices, but they are perturbed by the same noise, $\boldsymbol\nu$. Can I say that the distributions of $\mathbf{H}_1$ and $\mathbf{H}_2$ are independent of each other. i.e. is $p\left(\mathbf{H}_1\mathbf{H}_2\right)=p\left(\mathbf{H}_1\right)p\left(\mathbf{H}_2\right)$?

Answer 1

No, you cannot. As $H_2 = H_1 - (Y - X)$, $H_2$ is the difference of $H_1$ and something deterministic, that is, given $H_1$, we know $H_2$, so $H_1$ and $H_2$ are dependent.