I recently created a related topic about the square root matrix, in case you'd like to refer to that one.
Here's what we want: Consider the matrix $\Omega=E(\mathbf{u}^{\top}\mathbf{u})$, where $\mathbf{u}$ is $n \times 1$, so that $\Omega$ is $n \times n$. Notice $\mathbf{u}^{\top}\mathbf{u}$ is symmetric and positive definite; therefore, $\Omega=E(\mathbf{u}^{\top}\mathbf{u})$ should also be symmetric and positive definite. Since it is symmetric and positive definite, therefore it is invertible, so $\Omega^{-1}$ exists. Moreover, $\Omega^{-1}$ is also symmetric and positive definite.
Next, we use the fact of the symmetry and positive definiteness of $\Omega^{-1}=\left[E(\mathbf{u}^{\top}\mathbf{u})\right]^{-1}$, to decompose it as \begin{equation} \Omega^{-1}=\Omega^{-1/2}\Omega^{-1/2} \end{equation} where the square root $\Omega^{-1/2}$ is also symmetric, and with the additional property that \begin{equation} \Omega^{-1/2}\Omega\Omega^{-1/2}=I \end{equation} It was this particular property that inspired my previous question.
My new question is this: what kind of matrix decomposition are we using here? From looking at the Wikipedia page for matrix decomposition, I think it is either
- Cholesky Decomposition
- Eigendecomposition
- Jordan Normal Form Decomposition
which one, however, I am not sure.
By the way: this matrix decomposition into square root matrices is an application used in Generalized Least Squares, and there seems to be a lot of misunderstanding among those using it. The reason I say that is because a quick Google search shows that each author refers to what they're doing here as something different.
Some authors say they're doing Cholesky Decomposition; others talk about eigenvalues. Some authors even decompose $\Omega^{-1}=\Psi\Psi^{\top}$ (with transposes!), which seems quite unnecessary to me, since I believe the square root should be symmetric, also. The author I am choosing to follow writes $\Omega^{-1}=\Omega^{-1/2}\Omega^{-1/2}$ (without transposes), and I think this is optimal, since again, I believe $\Omega^{-1/2}$ is symmetric (though I don't know how to prove that).
So, can anyone set this straight? What kind of decomposition are we actually using here for this problem? Thank you for your thoughts!