In a course taught by Morris Eaton on multivariate statistics that dealt mostly with the Wishart distribution, I learned this proposition: Suppose
$$ M = \begin{bmatrix} A & B \\\\ B^T & C \end{bmatrix}$$
$A \in \mathbb{R}^{p\times p}$, $C \in \mathbb{R}^{q\times q}$, $B \in \mathbb{R}^{p \times q}$, and $M$ is positive-definite. Let $$ f(A, B, C) = (A - BC^{-1}B^T,\ C,\ BC^{-1}). $$ (The first component is the "Schur complement" of $C$ relative to $M$.)
Then $f$ is onto the set $$ \left\{ p\times p \text{ positive-definite} \right\} \times \left\{ q\times q \text{ positive-definite} \right\} \times \left\{ p \times q \right\} $$ (unlike the mapping $M \mapsto (A,C,B)$, whose image seems to be a messy subset because of the third component). The proof that it's one-to-one and onto isn't very hard.
My questions are:
(1) What is the history of this result? Who introduced it? (I asked Morris Eaton some time later and he said only that he learned it in graduate school and didn't know its origin.)
(2) What is it used for besides its uses in the theory of Wishart matrices?
Two observations: If $W = \begin{bmatrix} X \\ Y \end{bmatrix} \in \mathbb{R}^{(p+q)\times 1}$ is a normally distributed random column vector with expected value $0$ and variance $E(WW^T )= M$, and $X \in \mathbb{R}^p$ and $Y \in \mathbb{R}^q$, then (1) the conditional variance $\text{var}(X \mid Y)$ is $A - BC^{-1}B^T$ and (2) the conditional expected value of $X$ given $Y$ is $BC^{-1}Y$. Those facts were never mentioned in Eaton's course as far as I recall.
