Editing Covariance matrix (section)

===Inverse of the covariance matrix===
The [[invertible matrix|inverse of this matrix]], <math>\operatorname{K}_{\mathbf{X}\mathbf{X}}^{-1}</math>, if it exists, is the inverse covariance matrix (or inverse concentration matrix{{dubious|reason=An inverse concentration gets smaller the more concentrated something is, not larger as it reasonably should be. Besides, we already call it "concentration matrix" later on in this sentence, which is kind of contradictory.|date=September 2024}}), also known as the ''[[precision matrix]]'' (or ''concentration matrix'').<ref>{{cite book |title=All of Statistics: A Concise Course in Statistical Inference |url=https://archive.org/details/springer_10.1007-978-0-387-21736-9 |first=Larry |last=Wasserman |year=2004 |publisher=Springer |isbn=0-387-40272-1}}</ref>

Just as the covariance matrix can be written as the rescaling of a correlation matrix by the marginal variances: 
<math display="block">\operatorname{cov}(\mathbf{X})
= \begin{bmatrix}
 \sigma_{x_1} & & & 0\\
 & \sigma_{x_2}\\
 & &  \ddots\\
 0 & & &  \sigma_{x_n}
\end{bmatrix}
\begin{bmatrix}
 1 & \rho_{x_1, x_2} & \cdots & \rho_{x_1, x_n}\\
 \rho_{x_2, x_1} & 1 & \cdots & \rho_{x_2, x_n}\\
 \vdots & \vdots & \ddots & \vdots\\
 \rho_{x_n, x_1} & \rho_{x_n, x_2} & \cdots & 1\\
\end{bmatrix}
\begin{bmatrix}
 \sigma_{x_1} & & & 0\\
 & \sigma_{x_2}\\
 & &  \ddots\\
 0 & & &  \sigma_{x_n}
\end{bmatrix}</math>

So, using the idea of [[partial correlation]], and partial variance, the inverse covariance matrix can be expressed analogously:
<math display="block">\operatorname{cov}(\mathbf{X})^{-1}
= \begin{bmatrix}
 \frac{1}{\sigma_{x_1|x_2...}} & & & 0\\
 & \frac{1}{\sigma_{x_2|x_1,x_3...}}\\
 & &  \ddots\\
 0 & & &  \frac{1}{\sigma_{x_n|x_1...x_{n-1}}}
\end{bmatrix}
\begin{bmatrix}
 1 & -\rho_{x_1, x_2\mid x_3...} & \cdots & -\rho_{x_1, x_n\mid x_2...x_{n-1}}\\
 -\rho_{x_2, x_1\mid x_3...} & 1 & \cdots & -\rho_{x_2, x_n\mid x_1,x_3...x_{n-1}}\\
 \vdots & \vdots & \ddots & \vdots\\
 -\rho_{x_n, x_1\mid x_2...x_{n-1}} & -\rho_{x_n, x_2\mid x_1,x_3...x_{n-1}} & \cdots & 1\\
\end{bmatrix}
\begin{bmatrix}
 \frac{1}{\sigma_{x_1|x_2...}} & & & 0\\
 & \frac{1}{\sigma_{x_2|x_1,x_3...}}\\
 & &  \ddots\\
 0 & & &  \frac{1}{\sigma_{x_n|x_1...x_{n-1}}}
\end{bmatrix}</math>
This duality motivates a number of other dualities between marginalizing and conditioning for Gaussian random variables.