QR decomposition or SVD of the sum of matrices

Question

Apologies if this sounds like an uninformed question but I was wondering if there are theoretical results that talk about the following problem:

Suppose we have two mxn (m>=n) matrices A and B and they can be written as the following via QR decomposition:

A = Q_AR_A
B = Q_BR_B

Is there a way we can get the QR decomposition of the matrix (A+B) without explicitly adding A and B together and by only using the individual QR decomposition of both A and B. Specifically, I want to know if there are theoretical results that either talk about the feasibility of this or if not, provide a justification of why it cannot be done. Also, can the same be said about the SVD of (A+B)?

Answer 1

Let $A,B\in\mathbb{R}^{2\times 2}$ such that : $$ \mathbf{A}:=\begin{bmatrix}\varepsilon&0\\0&\varepsilon \end{bmatrix}\qquad\text{and}\qquad\mathbf{B}:=\begin{bmatrix}\frac{1}{\varepsilon}&0\\0&\frac{1}{\varepsilon}\end{bmatrix} $$ where $\varepsilon<<<1$. The QR decomposition of $A+B$ is : $$ \mathbf{Q}=\begin{bmatrix}1&0\\0&1\end{bmatrix} \qquad\text{and}\qquad\mathbf{R}=\begin{bmatrix}\varepsilon+\frac{1}{\varepsilon}&0\\0&\varepsilon+\frac{1}{\varepsilon}\end{bmatrix} $$ Unfortunately, $\displaystyle\frac{1}{\varepsilon}$ will absorb $\varepsilon$ which means $R=B$ which is against theory and this occurs when dealing with double float finite precision systems. Therefore, the QR decomposition or even SVD have relatively no special effect in optimizing matrix-matrix operations.

Here is a MATLAB demonstration on the command line:

e=1e-15;
A=e*eye(2);
B=(1/e)*eye(2);
[Q,R]=qr(A+B)