# $n,m$ symmetry in the determinants of block tridiagonal Toeplitz matrices

This question is related to [Determinant of block tridiagonal Toeplitz matrices] (Determinant of block tridiagonal Toeplitz matrices).

$$n\times n$$ block tridiagonal matrix $$A_{nm}$$ constructed from $$m\times m$$ blocks $$J_m,I_m$$,

\begin{align} A_{nm} &= \begin{bmatrix} J_n & I_n & 0 & \cdots & \cdots & 0 \\ I_m & J_m & I_m & 0 & \cdots & 0 \\ 0 & I_m & J_m & I_m & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 \\ 0 & \cdots & \cdots & I_m & J_m & I_m \\ 0 & \cdots & \cdots & \cdots & I_m & J_m \end{bmatrix} _{n\times n} , \end{align}

where $$I_m$$ is $$m\times m$$ identity matrix, and $$J_m$$ is a tridiagonal matrix with ones on the three diagonals.

\begin{align} \end{align}

The question: is this a known property or is there a simple way to prove that

\begin{align} \det(A_{nm})&=\det(A_{mn}) \tag{1}\label{1} \end{align}

for all $$n,m\in\mathbb N$$?

Statement \eqref{1} agrees for small $$n,m$$, for example this is a $$9\times9$$ matrix with elements $$m_{ij}=\det(A_{ij}),\ i,j=1,\dots,9$$:

\begin{align} M_{9} &= \begin{bmatrix} 1& 0& -1& -1& 0& 1& 1& 0& -1 \\ 0& -3& 0& 5& 0& -7& 0& 9& 0 \\ -1& 0& -7& -9& 0& -7& 119& 0& 369 \\ -1& 5& -9& 0& 55& 29& 279& -95& 0 \\ 0& 0& 0& 55& 0& -1183& 0& 0& 0 \\ 1& -7& -7& 29& -1183& 2197& -791& 28672& -165271 \\ 1& 0& 119& 279& 0& -791& -34391& 0& -2733921 \\ 0& 9& 0& -95& 0& 28672& 0& -4002939& 0 \\ -1& 0& 369& 0& 0& -165271& -2733921& 0& 0 \end{bmatrix} , \end{align}

where the first and second row/column shows an easy pattern, but the other (as well as a main diagonal) are not recognized as a known integer sequence.

For some reason, the choice of $$n=5$$ of $$m=5$$ result in seven zeros of nine.

Obviously, matrices $$A_{nm}$$ and $$A_{mn}$$ are of the same size, $$mn\times mn$$ elements and it's easy to show that they have the same number of $$1$$s.

As it is shown in this answer, $$\det(A_{nm})$$ boils down to the determinant of the $$m\times m$$ matrix, for example

\begin{align} \det(43)&= \left| \begin{matrix} 4&6&5 \\ 6&9&6\\ 5&6&4 \end{matrix} \right|=-9 ,\\ \det(34)&= \left| \begin{matrix} 2&3&3&1\\ 3&5&4&3\\ 3&4&5&3\\ 1&3&3&2 \end{matrix} \right|=-9 . \end{align}

Eigenvalues are given exactly by $$f(\theta_1,\theta_2)=1+2\cos(\theta_1)+2\cos(\theta_2)$$ where $$\theta_1=i\pi/(m+1)$$ $$i=1,\ldots,m$$ and $$\theta_2=j\pi/(n+1)$$ $$j=1,\ldots,n$$.