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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.

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@Michael: sorry about the edit. I tried to look at why the Math codes won't parse, but couldn't see how to fix it. – Willie Wong Dec 12 '10 at 12:44
Would it be possible to choose a title that contained some information about the topic of the question? – JBL Dec 12 '10 at 21:05
One could add a third "observation": If $M$ is a Wishart-distributed matrix, then so is the Schur complement. That was of some importance in Eaton's course. – Michael Hardy Dec 12 '10 at 23:15

That source of all wisdom, Wikipedia, contains the proof of your assertion (without saying so) -- look in the "Background" section of the Schur's complement article -- in more generality. Since I assume that was Schur's argument in the first place, I would further assume that the origin is Schur (but look at the Wikipedia article -- they claim some prehistory, and refer to Fuzhen Zhang's book on the subject.

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I'm not sure which of my assertions you're referring to. The one about surjectivity (and injectivity too) is proved easily by thinking about the inverse function of $f$. The ones about conditional expected values and conditional variances have standard proofs (which you're more likely to learn in a statistics course than in a probability course, since as a practical matter statisticians can't live without that stuff). However, your mention of Fuzhen Zhang's book may be the answer to my question about applications, and possibly also to my question about the history of the thing. Thank you. – Michael Hardy Dec 12 '10 at 21:34
Sorry, I was talking about injectivity and surjectivity (you get from one triple to another by multiplying by an invertible matrix...) – Igor Rivin Dec 13 '10 at 15:17

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