Can someone help me for the following question please?
Imagine you have a random vector $\mathbf{Y} = (Y_1, \cdots, Y_n)$ that takes value in $\mathbb{R}^n$ and that $Y_1,\cdots, Y_n$ are independent. Furthermore imagine that a linear subspace $M_0 \subset \mathbb{R}^n$ of dimension $p <n$ is given. Let $H_0 \in \mathbb{R}^{n\times n}$ (of rank p) be the projection matrix on $M_0$. Let also $Q \in \mathbb{R}^{p\times n}$ and $P \in \mathbb{R}^{(n-p)\times n}$ be two matrices such that:
- $PP^\top = I_{n-p}$ and $P^\top P = I_n - H_0$
- $QQ^\top = I_{p}$ and $Q^\top Q = H_0$
Where $I_m$ designates the identity matrix in $\mathbb{R}^{m\times m}$. Then the question is: Are $Q\mathbf{Y}$ and $P\mathbf{Y}$ independent?
I read somewhere in a master thesis that this two variables are inependent, however i don't understand why it should be the case. My reasoning is that the two variables $Q\mathbf{Y}$ and $P\mathbf{Y}$ can contain linear combinations of $Y_1,\cdots,Y_n$, and so, they are not independent.
What do you think?
Finally i found how to do it!
First, let us denote $\boldsymbol \mu$ the mean of $\mathbf{Y}$. Secondly, we can rewrite $P\mathbf{Y} = PP^\top P \mathbf{Y} = P(I_n-H_0)\mathbf{Y}$ and $Q\mathbf{Y} = QQ^\top Q \mathbf{Y} = QH_0\mathbf{Y}$. And, so, we have that $\mathbb{E}(P\mathbf{Y}) = P(I_n-H_0)\boldsymbol \mu$ and $\mathbb{E}(Q\mathbf{Y}) = Q(I_n-H_0)\boldsymbol \mu$.
Finally, we have that the covariance of $P\mathbf{Y}$ and $Q\mathbf{Y}$ is $\boldsymbol 0$:
\begin{align*} COV(P\mathbf{Y},Q\mathbf{Y}) &= \mathbb{E}[(P\mathbf{Y} - \mathbb{E}(P\mathbf{Y}))(Q\mathbf{Y} - \mathbb{E}(Q\mathbf{Y}))^\top] \\ & = \mathbb{E}[P(I_n-H_0)||\mathbf{Y}-\boldsymbol \mu||^2H_0 Q] \\ & = \boldsymbol 0. \end{align*}