Problem
=======


I was wondering if there are any theoretical results that tackle the following problem:



> 
> Construct the following matrices $\mathbf{\mathcal{S}\_{1}},\mathbf{\mathcal{S}\_{2}},\ldots,\mathbf{\mathcal{S}\_{p}}$ all of size $m\times n$ with $m\geqslant n$. The goal is to compute:
> $$
> \mathbf{\mathcal{S}}=\mathcal{S}\_{1}+\mathcal{S}\_{2}+\ldots+\mathcal{S}\_{p}
> $$
> not explicitly
> 
> 
> 


My goal is to add up these matrices not explicitly but rather by performing QR or SVD on each individual matrix and attempting to take advantage of the properties of the (assuming QR decomposition) orthonormal matrices $\mathbf{Q\_{\mathcal{S}\_{1}}},\mathbf{Q\_{\mathcal{S}\_{2}}},\ldots,\mathbf{Q\_{\mathcal{S}\_{p}}}$ and the upper triangular matrices $\mathbf{R\_{\mathcal{S}\_{1}}},\mathbf{R\_{\mathcal{S}\_{2}}},\ldots,\mathbf{R\_{\mathcal{S}\_{p}}}$ somehow (not by literally adding each pair together). Therefore, are there any theoretical result for matrix-matrix addition using matrix decomposition for the sake of computational feasibility and can the same be said using SVD?


Motivation
==========


Many matrix decomposition allow computations to be simpler by exploiting their special properties. The most important aspect of handling matrices numerically is to reduce numerical errors obtained due to finite precision on computers for instance addition operation of matrices are sometimes subject to the issue of absorption. Therefore, it might/might not be useful to look at the properties of matrix decompositions to tackle this issue which is why I am asking this question and I would be grateful for any comment/answer :)



Using QR or SVD to sum up finite number of matrices
If your matrices happen to be low-rank, you might want to consider a simultaneous low-rank approximation of your matrices (as in [Inoue and Urahama - Equivalence of Non-Iterative Algorithms for Simultaneous Low Rank Approximations of Matrices](https://doi.org/10.1109/CVPR.2006.112)), i.e. you compute
$$
\min\_{A,M\_i,B} \sum\_{i=1}^p \lVert S\_i - A M\_i B \rVert\_F^2,
$$
where $M\_i$ are potentially smaller than $S\_i$. You could then compute $$ S \approx A \bigl( \sum\_{i=1}^p M\_i \bigr) B.$$


I am however not aware of any result on the error of $S$. I would suspect that the error is equal to the sum of the individual low-rank approximation errors.


