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I am currently reading the book titled "Generalized Least Squares" by Takaeki Kariya and Hiroshi Kurata. In one section, a General linear regression model of the form \begin{equation} y=X\beta+\varepsilon \end{equation} where $y$ is an $n\times 1$ vector, $X$ is an $n\times k$ known matrix of full rank and $\varepsilon$ is an $n\times 1$ random vector with mean $0$ and covariance matrix $\Omega$, such that

\begin{equation} \mathbb{E}(\varepsilon)=0,\quad Cov(\varepsilon)=\mathbb{E}(\varepsilon\varepsilon')=\Omega\in S(n) \end{equation} where $S(n)$ denotes the set of $n\times n$ positive definite matrices. Moreover, $\Omega$ is unknown and is fomrulated as a function of an unknown but estimable parameter $\theta$ \begin{equation} \Omega=\Omega(\theta) \end{equation} A family of models they consider has the following covariance structure: \begin{equation} \Omega=\sigma^2\Sigma(\theta) \end{equation} with \begin{equation} \Sigma(\theta)^{-1}=I_n+\lambda_n(\theta)C\quad \theta\in\Theta\subset \mathbb{R} \end{equation} where $C$ is an $n\times n$ known symmetric matrix, $\lambda=\lambda_n=\lambda_n(\theta)$ is a continuous real valued function on $\Theta$, and the matrix $\Sigma(\theta)$ is positive definite for any $\theta\in\Theta$.

I follow everything clearly up until the presentation of the precision matrix. I would appreciate clarifying this for me, by the means of an example.

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Example: let $\lambda_n(\theta) = \theta^3$ and let $C = \begin{bmatrix}1 & 1 \\ 1 & 1 \end{bmatrix}$. Then

$$\Sigma(\theta) = (I_n + \lambda_n(\theta) C)^{-1} = \left(\begin{bmatrix}1 \\ & 1 \end{bmatrix} + \theta^3 \begin{bmatrix}1 & 1 \\ 1 & 1 \end{bmatrix} \right)^{-1} = \begin{bmatrix}\theta^3 + 1 & \theta^3 \\ \theta^3 & \theta^3 + 1\end{bmatrix}^{-1}.$$

The class of covariance matrices would be $\sigma^2 \Sigma(\theta)$ for $\theta \in \Theta$ where $\Sigma(\theta)$ has the above form and is positive definite.

For different choices of the function $\lambda_n(\cdot)$ and the fixed matrix $C$, you will get different sets of covariance matrices.

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