# Derivation of the matrix condition number of linear equations

When I was reading the derivation of the matrix condition number for linear equations on Wikipedia, $$||\textbf{b}||$$ can be directly replaced by $$||\textbf{Ax}||$$ as the non-singular square matrix is considered in the picture (which is the derivation on Wikipedia page).

Derivation of the condition number in Wikipedia

But, if we consider a rectangular matrix $$\textbf{A}$$, I don't think we can just substitute $$\textbf{b}$$ with $$\textbf{A}\textbf{x}$$. Then how we derive to achieve $$cond(\textbf{A})=||\textbf{A}||\cdot||\textbf{A}^{\dagger}||$$, where $${\dagger}$$ represents the pseudo inverse here?

• You may want to review your question... why would you replace $b$ by $A^{\dagger} x$? in the square matrix case this would translate as replacing $b$ with $A^{-1} x$, which does not make sense. Apr 20 at 12:26
• Sorry, I made a typo here. It's actually $\textbf{Ax}$. Apr 20 at 12:31

If two vectors are the same, so are their norms. Hence, if $$b = A x$$, it is always true that $$\|b\| = \|A x\|$$, regardless of the nature of the operator $$\mathcal A x := A x$$.
The condition number is much more delicate, in the sense that when a system has multiple solutions, it is not immediate how to measure sensitivity of the solution to perturbations in $$b$$. I recommend reading the paper: