How to Change Summation Expression $\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i$ into Matrix Expression Let $\mathbf{X}_i$ be a $G \times K$ matrix, and suppose are $i=1,...,N$ of these matrices. Note that
\begin{align}
\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{X}_i
&=
\begin{bmatrix}
\mathbf{X}_1^{\top} & \mathbf{X}_2^{\top} & \cdots & \mathbf{X}_N^{\top}
\end{bmatrix}
\begin{bmatrix}
\mathbf{X}_1 \\
\mathbf{X}_2 \\
\vdots \\
\mathbf{X}_N
\end{bmatrix} \\
&=\begin{bmatrix}
\mathbf{X}_1 \\
\mathbf{X}_2 \\
\vdots \\
\mathbf{X}_N
\end{bmatrix}^{\top}
\begin{bmatrix}
\mathbf{X}_1 \\
\mathbf{X}_2 \\
\vdots \\
\mathbf{X}_N
\end{bmatrix} \\
&=\mathbf{X}_{\bullet}^{\top}\mathbf{X}_{\bullet}
\end{align}
where $\mathbf{X}_{\bullet}$ is an $NG \times K$ matrix whose $i^{th}$ row is $\mathbf{X}_i$.
I'd like to do something similar with this summation:
\begin{equation}
\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i
\end{equation}
where $\mathbf{\Omega}$ is some arbitrary symmetric $G \times G$ matrix, say for illustration
\begin{equation}
\mathbf{\Omega}
=\begin{bmatrix}
g_{11} & g_{12} & \cdots & g_{1G} \\
g_{12} & g_{22} & \cdots & g_{2G} \\
\vdots & \vdots & \ddots & \vdots \\
g_{1G} & g_{2G} & \cdots & g_{GG}
\end{bmatrix}
\end{equation}
This problem is not so simple as the first one, because now I have some $\mathbf{\Omega}^{-1}$ matrix in the middle of the sum. However, I think it can still be expressed in matrix notation. 
For those curious, this matrix algebra is behind the theory of Generalized Least Squares. Thanks very much for your help!
Edit I think I have figured it out. I have posted an answer below -- please let me know if you think it's correct!

Let's try this.
\begin{align}
\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i&=\mathbf{X}_1\mathbf{\Omega}^{-1}\mathbf{X}_1+\mathbf{X}_2\mathbf{\Omega}^{-1}\mathbf{X}_2+\cdots+\mathbf{X}_N\mathbf{\Omega}^{-1}\mathbf{X}_N \\
&=\begin{bmatrix}
\mathbf{X}_1^{\top} & \mathbf{X}_2^{\top} & \cdots & \mathbf{X}_N^{\top}
\end{bmatrix}
\begin{bmatrix}
\mathbf{\Omega}^{-1}\mathbf{X}_1 \\
\mathbf{\Omega}^{-1}\mathbf{X}_2 \\
\vdots \\
\mathbf{\Omega}^{-1}\mathbf{X}_N
\end{bmatrix} \\
&=\begin{bmatrix}
\mathbf{X}_1^{\top} & \mathbf{X}_2^{\top} & \cdots & \mathbf{X}_N^{\top}
\end{bmatrix}
\begin{bmatrix}
\mathbf{\Omega}^{-1} & \mathbf{0} & \cdots & \mathbf{0} \\
\mathbf{0} & \mathbf{\Omega}^{-1} & \cdots & \mathbf{0} \\
\vdots & \vdots & \ddots & \vdots \\
\mathbf{0} & \mathbf{0} & \cdots & \mathbf{\Omega}^{-1}
\end{bmatrix}
\begin{bmatrix}
\mathbf{X}_1 \\
\mathbf{X}_2 \\
\vdots \\
\mathbf{X}_N
\end{bmatrix} \\
&=\begin{bmatrix}
\mathbf{X}_1 \\
\mathbf{X}_2 \\
\vdots \\
\mathbf{X}_N
\end{bmatrix}^{\top}
\begin{bmatrix}
\mathbf{\Omega}^{-1} & \mathbf{0} & \cdots & \mathbf{0} \\
\mathbf{0} & \mathbf{\Omega}^{-1} & \cdots & \mathbf{0} \\
\vdots & \vdots & \ddots & \vdots \\
\mathbf{0} & \mathbf{0} & \cdots & \mathbf{\Omega}^{-1}
\end{bmatrix}
\begin{bmatrix}
\mathbf{X}_1 \\
\mathbf{X}_2 \\
\vdots \\
\mathbf{X}_N
\end{bmatrix} \\
&=\mathbf{X}_\bullet^{\top}\left(\mathbf{I}_N \otimes \mathbf{\Omega}^{-1}\right)\mathbf{X}_\bullet
\end{align}
where "$\otimes$" denotes the Kronecker Product.
