# Alternate definition of condition number

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!

• 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? Commented Jun 9, 2023 at 21:01
• 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 Commented Jun 9, 2023 at 21:09
• @whpowell96 The definition in the book is fine. It has norms around the quantities so it is not dividing vector with vector: $\frac{\|A\delta x\|}{\|\delta x\|} / \frac{\|Ax\|}{\|x\|}$ Commented Jun 9, 2023 at 21:22
• I am aware. I figured norms were implied and answered his question. Commented Jun 9, 2023 at 22:17

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 $$(|x|)_i = |x_i|$$.