Is there a formula to compute the determinant of a structurally symmetric $n$-banded matrix? I am specifically interested in the 5-banded matrix:


$$ \left[\begin{matrix}
c\_{0} & s\_{0} & 0 & w\_{0} & 0 & 0 & 0 & 0 & 0\\
n\_{0} & c\_{1} & s\_{1} & 0 & w\_{1} & 0 & 0 & 0 & 0\\
0 & n\_{1} & c\_{2} & 0 & 0 & w\_{2} & 0 & 0 & 0\\
e\_{0} & 0 & 0 & c\_{3} & s\_{3} & 0 & w\_{3} & 0 & 0\\
0 & e\_{1} & 0 & n\_{3} & c\_{4} & s\_{4} & 0 & w\_{4} & 0 & \dots \\
0 & 0 & e\_{2} & 0 & n\_{4} & c\_{5} & 0 & 0 & w\_{5}\\
0 & 0 & 0 & e\_{3} & 0 & 0 & c\_{6} & s\_{6} & 0\\
0 & 0 & 0 & 0 & e\_{4} & 0 & n\_{6} & c\_{7} & s\_{7}\\
0 & 0 & 0 & 0 & 0 & e\_{5} & 0 & n\_{7} & c\_{8} \\
& & & & \vdots & & & & &
\ddots\end{matrix}\right] $$


which is defined by the five sequences:


$$ \left\lbrace w\_0, w\_1, \cdots \right\rbrace, \left\lbrace s\_0, s\_1, \cdots \right\rbrace, \left\lbrace c\_0, c\_1, \cdots \right\rbrace, \left\lbrace n\_0, n\_1, \cdots \right\rbrace, \left\lbrace e\_0, e\_1, \cdots \right\rbrace $$


All other elements are zero.


It's not clear to me whether formulas for tridiagonal matrices can be extended straightforwardly to compute the determinant of the above matrix.


Notes:


* In my special case some terms $ n\_2 = n\_5 = n\_8 = \cdots = 0 $, $ s\_0 = s\_3 = s\_6 = \cdots = 0 $, but I'm interested in the solution for arbitrary sequences $w\_i$, $s\_i$, $c\_i$, $n\_i$, $e\_i$.
* All off-diagonal elements are nonnegative, and all diagonal elements are negative.



Determinant of structurally symmetric $n$-banded matrix?
This determinant can be calculated using the block transfer matrix method of Molinari (Linear Algebra and its Applications 429 (2008) 2221-2226, <https://arxiv.org/abs/0712.0681>), if $w\_i \neq 0$. You have $3 \times 3$ blocks that read, after renumbering,
$$
\mathbf A\_k = 
\begin{pmatrix}
c\_{k,0} & s\_{k,0} & 0 \\
n\_{k,0} & c\_{k,1} & s\_{k,1} \\
0 & n\_{k,1} & c\_{k,2} 
\end{pmatrix}, 
\,\,
\mathbf B\_k = 
\begin{pmatrix}
w\_{k,0} & 0 & 0 \\
0 & w\_{k,1} & 0 \\
s\_{k,2} & 0 & w\_{k,2} 
\end{pmatrix}, 
\,\,
\mathbf C\_k = 
\begin{pmatrix}
e\_{k,0} & 0 & n\_{k,2} \\
0 & e\_{k,1} & 0 \\
0 & 0 & e\_{k,2} 
\end{pmatrix}.
$$


Defining the block transfer matrix
$$
\mathbf T\_k = 
\begin{bmatrix}
-\mathbf B\_k^{-1} \mathbf A\_k & -\mathbf B\_k^{-1} \mathbf C\_k \\
 \mathbf 1 & \mathbf 0
\end{bmatrix},
$$
the determinant $\Delta$ of the $K \times K$ block matrix (with $3 \times 3$ blocks) above then reads
$$
\Delta = 
\det \langle \mathbf 1,\mathbf 0| \mathbf T\_K \mathbf T\_{K-1} \cdots \mathbf T\_1
|\mathbf 1,\mathbf 0\rangle 
\prod\_{k=1}^{K} \det(-\mathbf B\_{k}) .
$$
Note that I used bra-ket notation, where the $2 \times 1$ block ket vector $|\mathbf 1,\mathbf 0\rangle$ has total size $6 \times 3$, such that the bra-ket pair picks the (1,1)-block of the block matrix product. For a similar application, see 


*Hucht, Alfred*, [**The square lattice Ising model on the rectangle. I: Finite systems**](http://dx.doi.org/10.1088/1751-8121/aa5535), J. Phys. A, Math. Theor. 50, No. 6, Article ID 065201, 23 p. (2017). [ZBL1357.81154](https://zbmath.org/?q=an:1357.81154), <https://arxiv.org/abs/1609.01963> .


