# Condition number of $A^{-1}B$ where $A$ and $B$ are banded toeplitz matrices.

I'm looking at a filtering problem with feedback, which can be represented by the equation $A\underline{y} = B\underline{x}$, where $A$ and $B$ are lower triangular banded toeplitz matrices and $\underline{x}$ and $\underline{y}$ are vectors.

As an example, $A$ and $B$ are of the form

$$\begin{pmatrix} a_1 &0 &0 & \ldots &0\\ a_2 &a_1&0& \ldots&0 \\ a_3&a_2 &a_1& \ldots&0 \\ 0&a_3&a_2 & \ldots&0 \\ 0&0&a_3 & \ldots&0 \\ \cdot&\cdot&\cdot&\cdot&a_1\\ \end{pmatrix}$$

Here, the matrix $A$ (as well as $B$) is specified by the few non zero values in their first column.

I want to look at the condition number of $A^{-1}B$ (as a way of measuring how sensitive the system is to the A and B coefficients). Are there are any results in the literature that would help characterize the condition number in the limit of large matrices $A$ and $B$? Thanks in Advance.

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