Block diagonazling an 8by8 matrix

Question

I have an 8by 8 matrix and I want to transform it to a block diagonalized matrix which each block is a 4 by 4 matrix.

$$ \begin{bmatrix} m_0 +J_H & 0 & V & 0 & 0 & \Delta_t & 0 & 0 \\ 0 & -m_0-J_H & 0 & V & -\Delta_t & 0 & 0 & 0\\ V & 0 & -m_0+J_H & 0 & 0 & 0 & 0 & -\Delta_t \\ 0 & V & 0 & m_0-J_H & 0 & 0 & \Delta_t & 0\\ 0 & -\Delta_t & 0 & 0 & -m_0-J_H & 0 & -V & 0 \\ \Delta_t & 0 & 0 & 0 & 0 & m_0+J_H & 0 & -V \\ 0 & 0 & 0 & \Delta_t & -V & 0 & m_0 - J_H & 0 \\ 0 & 0 & -\Delta_t & 0 & 0 & -V & 0 & -m_0+J_H \\ \end{bmatrix} $$

Do you have any idea how I can take those delta terms to the diagonal blocks?! (I don't want to calculate eigenvalues)

Answer 1

Apply the orthogonal matrix $$P = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0& 0& 0\\ 0 & 0 & 1 & 0 & 0 & 0& 0& 0\\ 0 & 0 & 0 & 0 & 0 & 1& 0& 0 \\ 0 & 0 & 0 & 0 & 0 & 0& 0& 1 \\ 0 & 1 & 0 & 0 & 0 & 0& 0& 0 \\ 0 & 0 & 0 & 1 & 0 & 0& 0& 0 \\ 0 & 0 & 0 & 0 & 1 & 0& 0& 0 \\ 0 & 0 & 0 & 0 & 0 & 0& 1& 0 \end{pmatrix}$$ from both sides, i.e., calculate $$ P A P^T$$ with $A$ your matrix.