Condition Number of sum of Matrices

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Discussion Overview

The discussion revolves around the condition number of the sum of matrices, specifically exploring the hypothesis that the condition number of the sum of two matrices is less than or equal to the sum of their individual condition numbers. Participants examine this claim through mathematical reasoning and personal testing.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant proposes the hypothesis that cond(A+B) ≤ cond(A) + cond(B) based on tests conducted in Matlab.
  • Another participant provides a mathematical expression attempting to support the hypothesis but questions the validity of the assumption that (A+B)⁻¹ = A⁻¹ + B⁻¹.
  • Concerns are raised about whether certain terms can be omitted in the mathematical reasoning, specifically ||A⁻¹||⋅||B|| + ||A||⋅||B⁻¹|.
  • A later reply acknowledges the initial doubts and admits to deleting a previous post due to realizing the reasoning was incorrect.
  • One participant concludes that the hypothesis cond(A+B) ≤ cond(A) + cond(B) is not always true, indicating that the hypothesis is incorrect.

Areas of Agreement / Disagreement

Participants express disagreement regarding the validity of the hypothesis, with some questioning its correctness and others concluding that it is not always true. The discussion remains unresolved regarding a definitive proof or counterexample.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the properties of matrix inverses and the conditions under which the hypothesis might hold. The mathematical steps presented are not fully resolved.

Abbas
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As far as I know there is no explicit formulas but is this true? I've tested it in Matlab with random matrices and It seems true!
cond(A+B) =< cond(A) + cond(B)
Where can I find a proof for this hypothesis?
 
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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}

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?
 
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?

You're quite right. I had just deleted my post, since I realized it was not correct due to the very reasons you mention.
 
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.

I was looking for an answer since I post it here, cond(A+B) =< cond(A) + cond(B) is not always true. the hypothesis is wrong. Thanks BTW. :)
 

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