Editing Schur complement (section)

==Applications to probability theory and statistics==

Suppose the random column vectors ''X'', ''Y'' live in '''R'''<sup>''n''</sup> and '''R'''<sup>''m''</sup> respectively, and the vector (''X'', ''Y'') in '''R'''<sup>''n'' + ''m''</sup> has a [[multivariate normal distribution]] whose covariance is the symmetric positive-definite matrix

:<math>\Sigma = \left[\begin{matrix} A & B \\ B^\mathrm{T} & C\end{matrix}\right],</math>

where  <math display="inline">A \in \mathbb{R}^{n \times n}</math> is the covariance matrix of ''X'', <math display="inline">C \in \mathbb{R}^{m \times m}</math> is the covariance matrix of ''Y'' and <math display="inline">B \in \mathbb{R}^{n \times m}</math>  is the covariance matrix between ''X'' and ''Y''.

Then the [[Conditional variance|conditional covariance]] of ''X'' given ''Y'' is the Schur complement of ''C'' in <math display="inline">\Sigma</math>:<ref name="von Mises 1964">{{cite book  |title=Mathematical theory of probability and statistics |url=https://archive.org/details/mathematicaltheo0057vonm |url-access=registration |first=Richard |last=von Mises |year=1964|publisher=Academic Press| chapter=Chapter VIII.9.3|isbn=978-1483255385}}</ref>

:<math>\begin{align}
  \operatorname{Cov}(X \mid Y) &= A - BC^{-1}B^\mathrm{T} \\
    \operatorname{E}(X \mid Y) &= \operatorname{E}(X) + BC^{-1}(Y - \operatorname{E}(Y))
\end{align}</math>

If we take the matrix <math>\Sigma</math> above to be, not a covariance of a random vector, but a ''sample'' covariance, then it may have a [[Wishart distribution]].  In that case, the Schur complement of ''C'' in <math>\Sigma</math> also has a Wishart distribution.{{Citation needed|date=January 2014}}