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b is the least square estimator of the general linear model $Y=X{\beta}$ where X is a n x p design matrix and $ Y\in R^p, {\beta}\in R^n$.

I want to prove blue part below. I worked some as you can find in the image I attached. But I found $Var(Xb-X{\beta})$ part is wrong since X doesn't have inverse.

Is there any other way I can show $Var(Xb-X{\beta})={\sigma}^2 I$? If I cannot is there any other way to prove this problem?

Click here to see image

Horselook
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  • It would be better if you type out the image. Do you know the distribution of $b$, its mean vector and dispersion matrix? – StubbornAtom Oct 17 '21 at 11:05
  • This follows from the fact that if $Z\sim N_p(\mu,\Sigma)$ where $\Sigma$ is positive-definite, then $(Z-\mu)^T\Sigma^{-1}(Z-\mu)\sim \chi^2_p$. You can find a proof here and in its linked posts: https://math.stackexchange.com/q/3442471/321264. – StubbornAtom Oct 17 '21 at 11:17

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