I am studying a paper on edge detection and Toeplitz matrices. At a certain point they explain the Yule-Walker Algorithm and they say the following:
Consider symmetric toeplitz matrix $\mathbf{T}_k=\begin{pmatrix}1 & \gamma_1 & \gamma_2 & \dots & \gamma_{k-1}\\ \gamma_1 & 1 & \gamma_1 & \ddots & \gamma_{k-2}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \gamma_{k-1} & \gamma_{k-2} & \gamma_{k-3} & \dots & 1\end{pmatrix}$ , and system like $\mathbf{T}_k\mathbf{y}_k=-\mathbf{r}_k$ where $\mathbf{r}_k=(\gamma_1,...,\gamma_k)^t$ where $\gamma_k \in \mathbb{R}$ is any given real number.
We want to express the solution of $ \,\mathbf{T}_{k+1}\mathbf{y}_{k+1}=-\mathbf{r}_{k+1}$ using the $k$-th system. Writing $\mathbf{y}_{k+1}=(\mathbf{z}_k^t , a_k)^t$ and considering $\mathbf{J}_k \in \mathbb{R}^{k \times k} \, J_{ij}=\begin{cases} 1 & \text{if}\ j=k-i+1 \\ 0 & \text{otherwise} \end{cases}$, then the $(k+1)$th system can be rewritten as $$\begin{pmatrix}\mathbf{T}_k & \mathbf{J}_k\mathbf{r}_k \\ \mathbf{0}^t & 1\end{pmatrix} \begin{pmatrix}\mathbf{z}_k \\ a_k\end{pmatrix}=-\begin{pmatrix}\mathbf{r}_k \\ \gamma_{k+1}\end{pmatrix}$$ This way we obtain $$\mathbf{T}_k\mathbf{z}_k+a_k\mathbf{J}_k\mathbf{r}_k=-\mathbf{r}_k $$ $$\mathbf{r}_k^t\mathbf{J}_k\mathbf{z}_k=-\gamma_{k+1}$$ And until this everything is crystal clear, but now they say that these equations can be rewritten as follows
$$\begin{pmatrix}\mathbf{I}_k & \mathbf{J}_k\mathbf{y}_k \\ \mathbf{0}^t & 1\end{pmatrix}^t\begin{pmatrix}\mathbf{T}_k & \mathbf{J}_k\mathbf{r}_k \\ \mathbf{0}^t & 1\end{pmatrix}\begin{pmatrix}\mathbf{I}_k & \mathbf{J}_k\mathbf{y}_k \\ \mathbf{0}^t & 1\end{pmatrix}=\begin{pmatrix}\mathbf{T}_k & \mathbf{J}_\mathbf{0} \\ \mathbf{0}^t & \delta_k\end{pmatrix}$$ where $\delta_k=1+\mathbf{r}_k^t\mathbf{y}_k$
I tried to compute it and it gave me three equations, two of those were just one the transpose of the other but nothing. I tried to read it as a change of basis but nothing. I am sure this is kinda trivial and i'm missing something.