The discussion centers on the condition number of the sum of matrices, specifically the hypothesis that cond(A+B) ≤ cond(A) + cond(B). Initial tests in MATLAB suggested this inequality might hold, but further analysis revealed that the hypothesis is incorrect. The participants confirmed that the relationship does not always apply, particularly due to the properties of matrix inverses and norms. The conclusion is that the inequality cond(A+B) ≤ cond(A) + cond(B) is not universally valid.
PREREQUISITESMathematicians, data scientists, and engineers working with numerical methods and matrix computations will benefit from this discussion, particularly those interested in the stability and accuracy of matrix operations.
I like Serena said:Welcome to MHB, Abbas! :)
\begin{aligned}
\text{cond}(A+B)
&= ||(A+B)^{-1}|| \cdot ||A+B|| \\
&= ||A^{-1} + B^{-1}||\cdot ||A+B|| \\
&\le \Big(||A^{-1}||+||B^{-1}||\Big) \cdot \Big(||A||+||B||\Big) \\
&\le ||A^{-1}||\cdot||A|| + ||B^{-1}||\cdot||B|| \\
&= \text{cond}(A) + \text{cond}(B)
\end{aligned}
Abbas said:Thanks, but Are you sure if this is true?
I doubt (A+B)-1= A-1+B-1.
How about ||A-1||⋅||B||+||A||⋅||B-1|| ? can these terms be omitted?
I like Serena said:You're quite right. I had just deleted my post, since I realized it was not correct due to the very reasons you mention.