Given an $n\times n$ dense matrix $\mathbf{K}$, is there any decomposition that factorizes $\mathbf{K}$ into $m$ tridiagonal matrices such that $\mathbf{K} = \mathbf{B}_1 \, \mathbf{B}_2 \, \mathbf{B}_3 \cdots \mathbf{B}_m$ where each $\mathbf{B}_i$ has the following form?
$$\mathbf{B}_i = \begin{pmatrix} d_1 & u_1 \\ l_1 & d_2 & u_2 \\ & l_2 & \ddots & \ddots \\ & & \ddots & \ddots & u_{n-1} \\ & & & l_{n-1} & d_n \end{pmatrix}$$
Note that $m$ is not given. It could potentially be as small as $(n−1)$, as the product of $(n−1)$ tridiagonal matrices would create a fully dense matrix. However, I'm not sure that any arbitrary given $\mathbf{K}$ can be factorized with $(n−1)$ terms.
Background & Context
I'm an applied physicist working with topological optimization of 2D structures in wave physics. When performing such an optimization, we are trying to map the complex amplitude of $n$ input modes onto $n$ output modes. As such, the completed devices are performing parallel analog dense matrix multiplication. link
I would like to understand required size of the structure (in wavelengths) prior to optimization. Numerically, I find that there is a minimum size, below which the optimizer does not find a good solution.
My thinking is that the problem can be reduced by imagining that there is a small finite width $\Delta x$ which effectively exchanges energy between neighboring forward propagating modes and that this finite width could be found. A slab of this finite width would be equivalent to a tridiagonal matrix, mapping a set of input modes only onto their neighboring output modes. In order to achieve a dense matrix, one would need $m$ such energy exchanges between neighboring modes and the final size would be $m \cdot \Delta x$.
A related but slightly less abstract application would be for use in a mesh. Suppose that you have $n$ input cables each carrying a complex amplitude. On a given layer, each of these cables mixes with its nearest neighbors in some way, potentially using Mach-Zender Interferometers. A layer would again be performing the operation of multiplication by a tridiagonal matrix. How many layers, $m$, of local mixing would be necessary to achieve an arbitrary dense matrix and how would one find the values of the individual mixers?
However, after a bit of Googling, I wasn't able to find any factorization schemes which break a dense matrix into $m$ tridiagonal matrices.