Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Covariance ▷ Precision matrix and correlation matrix

Theorem: Let $X$ be a random vector. Then, the precision matrix of $X$ can be expressed in terms of its correlation matrix as follows

\[\label{eq:precmat-corrmat} \Lambda_{XX} = \mathrm{D}_X^{-1} \cdot \mathrm{C}_{XX}^{-1} \cdot \mathrm{D}_X^{-1} \; ,\]

where $\mathrm{D}_X^{-1}$ is a diagonal matrix with the inverse standard deviations of $X_1, \ldots, X_n$ as entries on the diagonal:

\[\label{eq:invdiagmat} \mathrm{D}_X^{-1} = \mathrm{diag}(1/\sigma_{X_1},\ldots,1/\sigma_{X_n}) = \begin{bmatrix} \frac{1}{\sigma_{X_1}} & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & \frac{1}{\sigma_{X_n}} \end{bmatrix} \; .\]

Proof: The precision matrix is defined as the inverse of the covariance matrix

\[\label{eq:precmat-covmat} \Lambda_{XX} = \Sigma_{XX}^{-1}\]

and the relation between covariance matrix and correlation matrix is given by

\[\label{eq:covmat-corrmat} \Sigma_{XX} = \mathrm{D}_X \cdot \mathrm{C}_{XX} \cdot \mathrm{D}_X\]

where

\[\label{eq:diagmat} \mathrm{D}_X = \mathrm{diag}(\sigma_{X_1},\ldots,\sigma_{X_n}) = \begin{bmatrix} \sigma_{X_1} & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & \sigma_{X_n} \end{bmatrix} \; .\]

Using the matrix product property

\[\label{eq:matprod-inv} \left(A \cdot B \cdot C\right)^{-1} = C^{-1} \cdot B^{-1} \cdot A^{-1}\]

and the diagonal matrix property

\[\label{eq:diagmat-inv} \mathrm{diag}(a_1,\ldots,a_n)^{-1} = \begin{bmatrix} a_1 & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & a_n \end{bmatrix}^{-1} = \begin{bmatrix} \frac{1}{a_1} & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & \frac{1}{a_n} \end{bmatrix} = \mathrm{diag}(1/a_1,\ldots,1/a_n) \; ,\]

we obtain

\[\label{eq:precmat-corrmat-qed} \begin{split} \Lambda_{XX} &\overset{\eqref{eq:precmat-covmat}}{=} \Sigma_{XX}^{-1} \\ &\overset{\eqref{eq:covmat-corrmat}}{=} \left( \mathrm{D}_X \cdot \mathrm{C}_{XX} \cdot \mathrm{D}_X \right)^{-1} \\ &\overset{\eqref{eq:matprod-inv}}{=} \mathrm{D}_X^{-1} \cdot \mathrm{C}_{XX}^{-1} \cdot \mathrm{D}_X^{-1} \\ &\overset{\eqref{eq:diagmat-inv}}{=} \begin{bmatrix} \frac{1}{\sigma_{X_1}} & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & \frac{1}{\sigma_{X_n}} \end{bmatrix} \cdot \mathrm{C}_{XX}^{-1} \cdot \begin{bmatrix} \frac{1}{\sigma_{X_1}} & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & \frac{1}{\sigma_{X_n}} \end{bmatrix} \end{split}\]

which conforms to equation \eqref{eq:precmat-corrmat}.

Sources:

Metadata: ID: P122 | shortcut: precmat-corrmat | author: JoramSoch | date: 2020-06-06, 06:28.