I'm seeking for a proof to the following formula notated in my Numerical Analysis over Linear Algebra textbook, since the proof is not given as in other cases of estimations and errors :
Let $x\neq0$ the solution of the system $Ax=b$, where $A$ is a $n\times n$ invertile matrix and $x,b$ are $n-$dimensional. Let $y$ be an approximation of $x$ and $r=Ay-b$ the corresponding residue. Then, for any given norm $\| \cdot \|$ over $\mathbb R^n$ and the corresponding physical norm, the following inequality holds : $$\frac{\|y-x\|}{\|x\|} \leq k(A)\frac{\|r\|}{\|b\|}$$ where $k(A)$ is the condition number of the matrix $A$.
You mean : $$ \frac{\|y-x\|}{\|x\|} \le k(A)\frac{\|r\|}{\|b\|} $$ Solution $$ Ax=b $$ $$ \|Ax\|=\|b\| $$ $$ \|b\| \le \|A\| \|x\| , x \neq 0 $$ $$ \frac{1}{\|x\|} \le \frac{\|A\|}{\|b\|} , (i) $$
Now$$ \|y-x\| = \|y \|- \| A^{-1} b \| = \| A^{-1}Ay-A^{-1}b \|= \| A^{-1}(Ay-b) \| = \| A^{-1}(r) \| \le \| A^{-1}\| \|(r)\| , (ii) $$ So (i),(ii) $\to$ $$ \frac{\|y-x\|}{\|x\|} \le \frac{\|A\|\|A^{-1}\|\|r\|}{\|b\|}=k(A)\frac{\|r\|}{\|b\|} $$