I am seeking an exact justification for the following property. Consider a symmetric band matrix with real diagonal and imaginary off-diagonal. $$\left(\begin{array}{ccccc} a_1 & ib_1 & 0 & \dots & 0 \\ ib_1 & a_2 & ib_2 & \ddots & \vdots \\ 0 & ib_2 & \ddots & \ddots & 0 \\ \vdots & \ddots & \ddots & a_{N-1} & ib_{N-1} \\ 0 & \dots & 0 & ib_{N-1} & a_N \end{array}\right)$$
Numerically I just found that the structure of its inverse has real numbers on every even-numbered (off-)diagonal while every odd-numbered off-diagonal is imaginary. $$\left(\begin{array}{cccccc} c_1 & id_1 & e_1 & if_1 && \text{etc.}\\ id_1 & c_2 & ic_2 & e_2 & \ddots & \\ e_1 & id_2 & c_3 & id_3 & \ddots & if_{N-3} \\ if_1 & e_2 & id_3 & c_4 & \ddots & e_{N-2} \\ & \ddots & \ddots & \ddots & \ddots & id_{N-1} \\ \text{etc.} && if_{N-3} & e_{N-2} & id_{N-1} & c_N \end{array}\right)$$ Of course, all $a_n, b_n, c_n, d_n, e_n, f_n \in \mathbb{R}$. I can see how this makes sense with their product necessarily being identity, but I cannot think of a rigorous explanation. Could someone help me out?