Let $\mathbf{Z}$ be the design matrix of a linear regression.
Where $\mathbf{Z}^{T}\mathbf{Z}$ is symmetric.
Set $\mathbf{V} = (\mathbf{Z'Z})^{\frac{1}{2}}(E(\hat{\beta}) - \beta)$
Where $\hat{\beta} , \beta$ is the true and estimated parameter vectors of a linear regression model.
Thus we can say that:
$Cov(\mathbf{V}) = (\mathbf{Z^{T}\:Z})^{\frac{1}{2}} Cov(\hat{{\bf{\beta}}}) ((\mathbf{Z^{T}\:Z})^{\frac{1}{2}})^{T} =$
$(\mathbf{Z^{T}\:Z})^{\frac{1}{2}}\sigma^{2}(\mathbf{Z^{T}\:Z})^{-1}(\mathbf{Z^{T}\:Z})^{\frac{1}{2}} = \sigma^{2}\mathbf{I}$
Why is this? and from what matrix algebra rule does it follow?