I am solving a problem, in the middle of which I reached the following symmetric block-tridiagonal matrix.
\begin{bmatrix} 0 & B & &0 \\ B^T & 0 & \ddots \\ &\ddots& \ddots&B \\ 0 & & B^T & 0 \end{bmatrix} Where $B$ is an invertible square matrix and size $0$ is equal to size $B$(matrix B is not necessarily normal). Can I make this matrix similar to the following tridiagonal block matrix as shown below?
\begin{bmatrix} 0 & C & &0 \\ C & 0 & \ddots \\ &\ddots& \ddots&C \\ 0 & & C & 0 \end{bmatrix}
in which size $C$ and $0$ equals $B$.