I am preparing a work on Cochran's theorem and I had two questions :
- First question :
Is there a link between these two statements and to what extent? Would it be redundant to prove each of them separately ?
Theorem $1$ (Cochran, algebraic) :
Let $E$ be a euclidean space of dimension $n$ and $ u_1, ..., u_p $ be symmetric endomorphisms on E.
Suppose that :
$(i)$ $\text{rk}(u_1) +\ ... \ + \text{rk}(u_p) = n$.
$(ii)$ $q_1(x) + \ ... \ + q_p (x) = \langle x,x \rangle\ $ where $q_k(x) = \langle u_k(x),x\rangle$.
Then $E = \bigoplus\limits_{1\le i \le p}^{\perp}\text{Im}(u_i)$. Besides, for all $i\in \{1,...,p\}$, $u_i$ is an orthogonal projector on its image.
Theorem $2$ (Cochran, probabilistic) :
Let $X \sim \mathcal{N}_d (0, I_d) $ be a gaussian vector and $ \bigoplus \limits_{1\le i \le p}^{\perp} E_i = \mathbb{R}^d $ with $\dim(E_i)= d_i$ for all $i\in \{1,...,p\}$. Then the orthogonal projections $\pi_{E_1} (X), ..., \pi_{E_p} (X) $ are independent gaussian vectors. Besides, for all $i \in \{1,...,p\}$ : $\Vert \pi_{E_i}(X)\Vert^2 \sim \chi^2(d_i)$.
- Second question :
Does anyone know of an application of these theorems, other than the statistical gaussian linear model tests ? Specifically is there a purely algebraic application to theorem $1$ ?
Thanks in advance.