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I am trying to understand if the condition number of a matrix $A$ can be defined as $\left(\frac{\Delta x}{x}\right) / \left(\frac{\Delta b}{b}\right)$ under the following conditions: $$ Ax = b, \qquad A(x+Δx) = b+Δb $$ I have seen this definition mentioned, but I am not sure if it is a valid definition for the condition number of a matrix. If this is indeed a valid definition, could you please provide a reference or source that supports this definition?

I am aware of the more common definition of the condition number for a square matrix A, which is defined as:

$Cond(A) = ||A|| ||A^{-1}||$

However, I am specifically interested in understanding if the $\left(\frac{\Delta x}{x}\right) / \left(\frac{\Delta b}{b}\right)$ definition is valid and, if so, under what circumstances it can be used.

Any help or guidance on this topic would be greatly appreciated. Thank you in advance!

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  • $\begingroup$ No, for the obvious reason that $\Delta x/x$ is an ill-defined quantity (and likewise throughout): how does one even define division by a vector? $\endgroup$
    – PrincessEev
    Jun 9 at 21:01
  • $\begingroup$ Chapter 12 of Trefethen and Bau defines condition numbers in a manner similar to this and uses it to derive the standard definition of the matrix condition number $\endgroup$
    – whpowell96
    Jun 9 at 21:09
  • $\begingroup$ @whpowell96 The definition in the book is fine. It has norms around the quantities so it is not dividing vector with vector: $\frac{\|A\delta x\|}{\|\delta x\|} / \frac{\|Ax\|}{\|x\|}$ $\endgroup$
    – balddraz
    Jun 9 at 21:22
  • $\begingroup$ I am aware. I figured norms were implied and answered his question. $\endgroup$
    – whpowell96
    Jun 9 at 22:17

1 Answer 1

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You are considering the function $F : \mathbb{R}^{n} \rightarrow \mathbb{R}^n$ given by $$F(b) = A^{-1} b. $$ What you seek is known as the componentwise relative condition number of the function $F$. I discuss the abstract definition of normwise and componentwise relative condition numbers of general functions in this answer and present formulas that can be used to compute the condition numbers in practice. In your case, the formula is $$\kappa_F^{cr}(b) = \left \| \frac{|A^{-1}||Ax|}{|x|} \right \|_\infty, \quad x = A^{-1}b. $$ The definition and the formula are meaningful provided that every component of $x$ is nonzero. The fraction represents componentwise division between the two vectors $|A^{-1}||Ax|$ and $|x|$. Finally, the vector $|x|$ is given by $(|x|)_i = |x_i|$.

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  • $\begingroup$ @RodrigodeAzevedo What have I done that you perceive as vandalism? $\endgroup$
    – Carl Christian
    Jun 26 at 8:43
  • $\begingroup$ @RodrigodeAzevedo In my experience, the tag matrices is used mainly for problems that assume exact arithmetic. That is not the case here. Upon inspection, I see that removing the tag matrices was a mistake as the tag description literally includes the phrase "any problem involving matrices". I removed the tag "linear-algebra" as this tag covers the curriculum covered in an introduction to linear algebra and these topics are not important here. I added numerical-methods and numerical-linear-algebra as these tags cover the problem and the solution. $\endgroup$
    – Carl Christian
    Jun 26 at 9:19
  • $\begingroup$ If there is room for 5 tags, what do you gain by removing them? I subscribe to tags. If you remove tags, how can I find questions? $\endgroup$
    – Rodrigo de Azevedo
    Jun 26 at 17:53

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