I am reading the following book: Data Assimilation: Methods, Algorithms, and Applications (https://epubs-siam-org.iclibezp1.cc.ic.ac.uk/doi/book/10.1137/1.9781611974546) and they have mentioned BLUE and I understand the general idea and proofs available here, as related to the Guass-Markov Theorem (https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem#:~:text=In%20statistics%2C%20the%20Gauss%E2%80%93Markov,have%20equal%20variances%20and%20expectation).
I am however having one issue in that I cannot see how they have arrived at the equation for $\hat{B}= (K^TC_N^{-1}K)^{-1}K^TC_N^{-1}$ in Theorem 3.27. When I try and solve such a problem I get the OLS solution, which is given in Remark 3.28, $\hat{x}_{BLUE} = (K^TK)^{-1}K^Tz$.
Would someone be so kind as to tell me/give me a hint on how they have arrived at $\hat{B}$? Image of the page in question is given below.
Thanks all!
Gauss-Markov theorem gives the BLUE estimator when:
The theorem proven does not assume the covariance of errors is zero, as is stated in the remark this case can be obtained by selecting $C=I\sigma^2$ as the correlation matrix. In the case where errors are correlated OLS is NOT BLUE, and we use a different estimator. We can describe the model as follows
$$y=X\beta+C^{1/2}\epsilon,$$
where $C$ is the covariance matrix and $\epsilon$ are uncorrelated. Hence:
$$\Leftrightarrow C^{-1/2}y=C^{-1/2}X\beta+\epsilon,$$
And now we have a model with uncorrelated errors, running OLS with respect to it is BLUE, in particular by minimizing the sum squares we get $$\hat \beta = ((C^{-1/2}X)'(C^{-1/2}X))^{-1}(C^{-1/2}X)'C^{-1/2}y$$
$$=(X'C^{-1}X)^{-1}X'C^{-1}y$$
Which is what the theorem shows is BLUE (not OLS), as should be.