I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in).


I have two sequences of matrices $Q\_{1},\ldots,Q\_{k}$ and $R\_{1},\ldots,R\_{k}$ in $\mathbb{R}^{n^{2}}$ and would like to bound $|| \prod\_{i=1}^{k} Q\_{i} - \prod\_{i=1}^{k} R\_{i} ||\_{\mathrm{HS}}$ in terms of the $k$ numbers $|| Q\_{i} - R\_{i} ||\_{\mathrm{HS}}$.


If we have $Q\_{i}, R\_{i} \in SO(n)$ for all $i$, it is easy to check that


$|| \prod\_{i=1}^{k} Q\_{i} - \prod\_{i=1}^{k} R\_{i} ||\_{\mathrm{HS}} \leq \sum\_{i=1}^{k} || Q\_{i} - R\_{i} ||\_{\mathrm{HS}}$.


That isn't true more generally (e.g. if the matrices don't all have norm 1, this must appear in the upper bound), and I am looking for a standard reference for any similar fact (especially if it includes this bound as a special case).


I am particularly interested in:


1. $k$ large, $|| Q\_{i} - R\_{i} ||\_{\mathrm{HS}}$ small.
2. The case where, for each $i$, either $Q\_{i}, R\_{i} \in SO(n)$ or $Q\_{i}, R\_{i}$ are in the associated Lie algebra and have the same (moderate) norm.



Bounds on Hilbert-Schmidt norm of difference of products of matrices
I am not sure that you can get anything better than the following triangle inequality bound:


$$\left\|\prod\_{i=1}^k Q\_i - \prod\_{i=1}^k R\_i\right\|\_{HS} \leq \sum\_{h=1}^k \left\|\prod\_{i=1}^{h-1}Q\_i\right\|\_2 \, \left\|Q\_h-R\_h\right\|\_{HS}\, \left\|\prod\_{i=h+1}^k R\_k\right\|\_2.$$


EDIT: I have converted the outer HS norms to (smaller) operator norms, using the fact that $\|{AB}\|\_{HS} \leq \|A\|\_{HS}\|B\|\_2$ and $\|{AB}\|\_{HS} \leq \|A\|\_{2}\|B\|\_{HS}$. Note that this bound reduces to the one you found in the case when the $Q\_i$ and $R\_i$ are orthogonal.


