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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!

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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, $$\mathbf w = \begin{bmatrix} 0_m \\ m \mathbf u \end{bmatrix} = \begin{bmatrix} 0 \mathbf u \\ m \mathbf u \end{bmatrix} = \begin{bmatrix} 0 \\ m \end{bmatrix} \otimes \mathbf u.$$

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