[proofplan] We prove the result directly by matrix algebra rather than reducing to the ordinary homoskedastic case. First we show that $X^\top \Omega^{-1}X$ is invertible and that the proposed weighted least squares estimator is unbiased. Then we characterize every linear estimator $AY$ that is unbiased for every $\beta \in \mathbb{R}^p$ by the constraint $AX = I_p$, where $I_p \in \mathbb{R}^{p \times p}$ denotes the identity matrix. Finally we write $A$ as the weighted least squares matrix plus an error matrix $B$ satisfying $BX = 0$, expand the covariance, and show that all cross terms vanish while the remaining term $B\Omega B^\top$ is [positive semidefinite](/page/Positive%20Semidefinite%20Matrix). [/proofplan] [step:Show that the weighted normal matrix is invertible] Since $\Omega$ is symmetric [positive definite](/page/Positive%20Definite%20Matrix), the standard inverse-positivity theorem for positive definite matrices implies that $\Omega^{-1}$ is symmetric positive definite. Define the matrix \begin{align*} M := X^\top \Omega^{-1}X \in \mathbb{R}^{p \times p}. \end{align*} For every nonzero vector $v \in \mathbb{R}^p$, the full column rank of $X$ gives $Xv \neq 0$. Therefore \begin{align*} v^\top Mv = v^\top X^\top \Omega^{-1}Xv = (Xv)^\top \Omega^{-1}(Xv) > 0. \end{align*} Thus $M$ is symmetric positive definite. By the standard criterion that a positive definite square matrix has no zero eigenvalue, $M$ is [invertible](/page/Invertible%20Matrix). [guided] The estimator contains the inverse of $X^\top \Omega^{-1}X$, so the first point is to prove that this inverse exists. Define \begin{align*} M := X^\top \Omega^{-1}X \in \mathbb{R}^{p \times p}. \end{align*} Because $\Omega$ is symmetric [positive definite](/page/Positive%20Definite%20Matrix), the standard inverse-positivity theorem for positive definite matrices gives that its inverse $\Omega^{-1}$ is also symmetric positive definite. Now take any nonzero vector $v \in \mathbb{R}^p$. Since $X$ has rank $p$, its columns are linearly independent, so $Xv \neq 0$. Applying positive definiteness of $\Omega^{-1}$ to the nonzero vector $Xv \in \mathbb{R}^n$ gives \begin{align*} v^\top Mv = v^\top X^\top \Omega^{-1}Xv = (Xv)^\top \Omega^{-1}(Xv) > 0. \end{align*} Therefore $M$ is symmetric positive definite. A symmetric positive definite square matrix has no zero eigenvalue, so it is [invertible](/page/Invertible%20Matrix). [/guided] [/step] [step:Verify that the weighted least squares estimator is linear and unbiased] Define \begin{align*} A_{\Omega} := (X^\top \Omega^{-1}X)^{-1}X^\top \Omega^{-1} \in \mathbb{R}^{p \times n}. \end{align*} Then $\hat{\beta}_{\Omega} = A_{\Omega}Y$, so $\hat{\beta}_{\Omega}$ is linear in $Y$. Moreover, \begin{align*} A_{\Omega}X = (X^\top \Omega^{-1}X)^{-1}X^\top \Omega^{-1}X = I_p. \end{align*} Here $I_p \in \mathbb{R}^{p \times p}$ is the identity matrix, as defined in the proof plan. Using $Y = X\beta + \varepsilon$ and $\mathbb{E}[\varepsilon] = 0$, we obtain \begin{align*} \mathbb{E}[\hat{\beta}_{\Omega}] = \mathbb{E}[A_{\Omega}Y] = A_{\Omega}X\beta + A_{\Omega}\mathbb{E}[\varepsilon] = \beta. \end{align*} Thus $\hat{\beta}_{\Omega}$ is an unbiased linear estimator of $\beta$. [/step] [step:Characterize all uniformly unbiased linear estimators by the equation $AX = I_p$] Let $A \in \mathbb{R}^{p \times n}$ be such that $AY$ is an [unbiased linear estimator](/page/Linear%20Unbiased%20Estimator) of $\beta$ for every parameter value $\beta \in \mathbb{R}^p$. Since \begin{align*} \mathbb{E}[AY] = A\mathbb{E}[Y] = AX\beta, \end{align*} this uniform unbiasedness condition is equivalent to \begin{align*} AX = I_p. \end{align*} Indeed, if $AX = I_p$, then $\mathbb{E}[AY] = \beta$ for every $\beta$. Conversely, if $AX\beta = \beta$ for every $\beta \in \mathbb{R}^p$, then $(AX - I_p)\beta = 0$ for every $\beta \in \mathbb{R}^p$, which implies $AX = I_p$. [guided] A linear estimator of $\beta$ has the form $AY$, where $A \in \mathbb{R}^{p \times n}$. In the Gauss-Markov comparison class, unbiasedness means unbiasedness for every possible value of the parameter $\beta \in \mathbb{R}^p$, not merely at one fixed parameter value. Its expectation is determined by the mean of $Y$. Since $Y = X\beta + \varepsilon$ and $\mathbb{E}[\varepsilon] = 0$, we have \begin{align*} \mathbb{E}[AY] = A\mathbb{E}[Y] = A(X\beta) = AX\beta. \end{align*} For $AY$ to be unbiased for $\beta$ for every possible parameter value $\beta \in \mathbb{R}^p$, we must have \begin{align*} AX\beta = \beta \end{align*} for every $\beta \in \mathbb{R}^p$. This condition is exactly \begin{align*} AX = I_p. \end{align*} The implication in the other direction is immediate: if $AX = I_p$, then \begin{align*} \mathbb{E}[AY] = AX\beta = I_p\beta = \beta. \end{align*} Thus the matrices producing unbiased linear estimators are precisely those satisfying $AX = I_p$. [/guided] [/step] [step:Decompose an arbitrary unbiased estimator into the weighted estimator plus an orthogonal error] Let $A \in \mathbb{R}^{p \times n}$ satisfy $AX = I_p$. Define \begin{align*} B := A - A_{\Omega} \in \mathbb{R}^{p \times n}. \end{align*} Since both $A$ and $A_{\Omega}$ satisfy the unbiasedness equation, we have \begin{align*} BX = (A - A_{\Omega})X = AX - A_{\Omega}X = I_p - I_p = 0. \end{align*} Therefore \begin{align*} A = A_{\Omega} + B \end{align*} with $BX = 0$. [guided] We now compare an arbitrary unbiased linear estimator $AY$ with the weighted least squares estimator $A_{\Omega}Y$. The natural object to measure their difference is the matrix \begin{align*} B := A - A_{\Omega} \in \mathbb{R}^{p \times n}. \end{align*} The key structural fact is that $B$ kills the columns of $X$. Indeed, the previous step gives $AX = I_p$, and the direct computation of $A_{\Omega}X$ gives $A_{\Omega}X = I_p$. Hence \begin{align*} BX = (A - A_{\Omega})X = AX - A_{\Omega}X = I_p - I_p = 0. \end{align*} This identity is the algebraic reason the covariance cross terms vanish in the next step. [/guided] [/step] [step:Expand the covariance difference and identify the positive semidefinite remainder] Since $\operatorname{Var}(Y) = \operatorname{Var}(\varepsilon) = \Omega$, covariance under a deterministic matrix gives \begin{align*} \operatorname{Var}(AY) = A\Omega A^\top, \qquad \operatorname{Var}(\hat{\beta}_{\Omega}) = A_{\Omega}\Omega A_{\Omega}^\top. \end{align*} Using $A = A_{\Omega} + B$, we expand: \begin{align*} A\Omega A^\top &= (A_{\Omega}+B)\Omega(A_{\Omega}+B)^\top \\ &= A_{\Omega}\Omega A_{\Omega}^\top + A_{\Omega}\Omega B^\top + B\Omega A_{\Omega}^\top + B\Omega B^\top. \end{align*} The first cross term vanishes because \begin{align*} A_{\Omega}\Omega B^\top &= (X^\top \Omega^{-1}X)^{-1}X^\top \Omega^{-1}\Omega B^\top \\ &= (X^\top \Omega^{-1}X)^{-1}X^\top B^\top \\ &= (X^\top \Omega^{-1}X)^{-1}(BX)^\top \\ &= 0. \end{align*} The second cross term is its transpose, so \begin{align*} B\Omega A_{\Omega}^\top = 0. \end{align*} Therefore \begin{align*} \operatorname{Var}(AY) - \operatorname{Var}(\hat{\beta}_{\Omega}) = B\Omega B^\top. \end{align*} For every $u \in \mathbb{R}^p$, \begin{align*} u^\top B\Omega B^\top u = (B^\top u)^\top \Omega (B^\top u) \geq 0, \end{align*} because $\Omega$ is positive definite. Hence $B\Omega B^\top$ is positive semidefinite. [guided] The [variance](/page/Variance) of a deterministic linear transformation of a random vector satisfies \begin{align*} \operatorname{Var}(AY) = A\operatorname{Var}(Y)A^\top. \end{align*} Here $\operatorname{Var}(Y) = \operatorname{Var}(X\beta+\varepsilon) = \operatorname{Var}(\varepsilon) = \Omega$, since $X\beta$ is deterministic. Therefore \begin{align*} \operatorname{Var}(AY) = A\Omega A^\top, \qquad \operatorname{Var}(\hat{\beta}_{\Omega}) = A_{\Omega}\Omega A_{\Omega}^\top. \end{align*} Substitute the decomposition $A = A_{\Omega}+B$: \begin{align*} A\Omega A^\top &= (A_{\Omega}+B)\Omega(A_{\Omega}+B)^\top \\ &= A_{\Omega}\Omega A_{\Omega}^\top + A_{\Omega}\Omega B^\top + B\Omega A_{\Omega}^\top + B\Omega B^\top. \end{align*} The important point is that the middle two terms vanish. For the first cross term, use the definition of $A_{\Omega}$ and the identity $\Omega^{-1}\Omega = I_n$, where $I_n \in \mathbb{R}^{n \times n}$ denotes the identity matrix: \begin{align*} A_{\Omega}\Omega B^\top &= (X^\top \Omega^{-1}X)^{-1}X^\top \Omega^{-1}\Omega B^\top \\ &= (X^\top \Omega^{-1}X)^{-1}X^\top B^\top \\ &= (X^\top \Omega^{-1}X)^{-1}(BX)^\top. \end{align*} But the previous step proved $BX = 0$, so \begin{align*} A_{\Omega}\Omega B^\top = 0. \end{align*} The other cross term is the transpose of this one: \begin{align*} B\Omega A_{\Omega}^\top = (A_{\Omega}\Omega B^\top)^\top = 0. \end{align*} Thus the entire covariance excess is \begin{align*} \operatorname{Var}(AY) - \operatorname{Var}(\hat{\beta}_{\Omega}) = B\Omega B^\top. \end{align*} Finally, to prove positive semidefiniteness, test against an arbitrary vector $u \in \mathbb{R}^p$: \begin{align*} u^\top B\Omega B^\top u = (B^\top u)^\top \Omega (B^\top u) \geq 0. \end{align*} The inequality follows from the positive definiteness of $\Omega$; it also covers the case $B^\top u = 0$, where the value is exactly $0$. Hence $B\Omega B^\top$ is positive semidefinite. [/guided] [/step] [step:Conclude the best linear unbiased property] For every linear estimator $AY$ that is unbiased for every $\beta \in \mathbb{R}^p$, we have shown that \begin{align*} \operatorname{Var}(AY) - \operatorname{Var}(\hat{\beta}_{\Omega}) = B\Omega B^\top \end{align*} for some $B \in \mathbb{R}^{p \times n}$, and this matrix is positive semidefinite. Therefore $\hat{\beta}_{\Omega}$ has covariance no larger than any other uniformly unbiased linear estimator in the Loewner order. Since $\hat{\beta}_{\Omega}$ is itself linear and unbiased for every $\beta \in \mathbb{R}^p$, it is the [best linear unbiased estimator](/page/Best%20Linear%20Unbiased%20Estimator) of $\beta$ in this comparison class. [/step]