Let $A\_0, \dots, A\_{n-1}$ be upper triangular matrices with ones on the diagonal. Let $B\_{n-1}, \dots, B\_0$ be of the same form.  

I am interested in bounding
$$|| A\_0 \dots A\_{n-1} B\_{n-1}^{-1} \dots B\_0^{-1}||$$
and in particular showing that this product is close to the identity when $A\_i$ and $B\_i$ are close (specifically, $||A\_i - B\_i|| \leq \delta \lambda^{- \min(i, n-i)}$ where $\lambda>1$).


In general, this will probably not be possible, but there may be conditions when there are interesting bounds.


The problem is, I don't really know anything about techniques for bounding products of matrices. There is the obvious multiplicativity of the matrix (i.e., operator) norm, but it is far from best possible in this case. For instance, it is easy to check that the norm can grow at most polynomially in $n$ if the norms of the $A\_i, B\_i$ are bounded above by some constant.


Hence:


**Question:** What techniques/tricks are there for bounding these kinds of products of matrices?


Unfortunately I don't know exactly which techniques would be useful, so I'd appreciate any pointers to relevant papers or books. 


I can at least explain how the problem is motivated. Namely, we have a Holder-continuous matrix-valued function $A: X \to GL\_m(\mathbb{R})$ for $X$ a compact metric space.  

Then the $A\_i$ will be the values $A(T^ix)$ and the $B\_i$ will be $A(T^ip)$ for $p$ close to $x$. The multiplicative ergodic theorem states that the norm products $||A\_0 \dots A\_{n-1}||$ grow like $e^{\lambda n}$ almost everywhere (in $x$) with respect to any invariant measure (where $\lambda$ is the maximal Lyapunov exponent), but in the case of upper triangular matrices, this is actually immediate---we even have polynomial growth if ones are on the diagonal (a corollary of the nilpotence of strictly upper triangular matrices). The multiplicative ergodic theorem and its variants were used in a paper that motivated the problem I'm working on, but it doesn't quite seem to help here (because we make slightly different assumptions), which is why I was curious about other techniques.



Techniques to bound products of upper triangular matrices and their inverses
The standard trick to quantify the joint continuity of a product operation such as $(A\_1,\ldots,A\_k) \to A\_1 \ldots A\_k$ (which is essentially what you are trying to do here) is to split a difference such as


$$ A\_1 \ldots A\_k - B\_1 \ldots B\_k $$


as the telescoping sum of $k$ expressions of the form


$$ A\_1 \ldots A\_{i-1} (A\_i - B\_i) B\_{i+1} \ldots B\_k$$


and then estimate each term of the latter separately by various standard inequalities, e.g.


$$ \|AB\|\_{op} \leq \|A\|\_{op} \|B\|\_{op}.$$


This will give some bound on the norm of the difference of the product in terms of the norms of the individual differences $A\_i - B\_i$.


A similar device will also control $A\_1 \ldots A\_{n-1} B\_{n-1}^{-1} \ldots B\_1^{-1} - I$.


