# Writing a matrix in an alternative form with a Kronecker product.

I need to express the matrix $$\begin{equation} \begin{bmatrix} I & A \\ A^T & O \\ \end{bmatrix} \end{equation}$$ where

$$A = \begin{bmatrix} m\textbf{u}^T\\ I_m\\ \end{bmatrix}, I_m = \begin{bmatrix} 1 & 0 & \dots & 0 \\ 0 & 1 & \ddots & \vdots\\ \vdots & \ddots &\ddots & 0\\ 0 & \dots & 0 & 1 \end{bmatrix} \in \mathbb R^{m \times m}, \mathbf u = \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix}\in\mathbb R^m$$

in the following form: $$\begin{equation} \begin{bmatrix} 1 & \textbf{w}^T \\ \textbf{w} & B \\ \end{bmatrix} \end{equation}$$ where $$B$$ needs to be the Kronecker product between a $$2 \times 2$$ matrix and $$I_m$$. I dont really understand how to do this, i have tried matching the corresponding entries of the two matrices to try and work it out that way but it doesn't seem to work. Any help would be great, thanks!

Split $$A$$ and $$I$$ accordingly in your larger matrix so as to visually distinguish the first column and row: \begin{align} \left[\begin{array}{c|c} I_{m+1} & A \\ \hline A^{\mathrm T} & O_m \end{array}\right] &= \left[\begin{array}{cc|c} 1 & 0^{\mathrm T}_m & m \mathbf u^{\mathrm T} \\ 0_m & I_m & I_m \\ \hline m \mathbf u & I_m & O_m \end{array}\right] \\ &= \left[\begin{array}{c|cc} 1 & 0^{\mathrm T}_m & m \mathbf u^{\mathrm T} \\ \hline 0_m & I_m & I_m \\ m \mathbf u & I_m & O_m \end{array}\right] . \end{align}
So your $$B$$ is $$B = \begin{bmatrix} I_m & I_m \\ I_m & O_m \end{bmatrix} = \begin{bmatrix} 1 I_m & 1 I_m \\ 1 I_m & 0 I_m \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} \otimes I_m.$$
Also, $$\mathbf w = \begin{bmatrix} 0_m \\ m \mathbf u \end{bmatrix} = \begin{bmatrix} 0 \mathbf u \\ m \mathbf u \end{bmatrix} = \begin{bmatrix} 0 \\ m \end{bmatrix} \otimes \mathbf u.$$