Let $ A \in \mathbb{R}^{n \times n}$ be a non-singular matrix.

Let $\hat A=A+\delta A, \ \hat x=x+\delta x, \ \text{and} \ \hat b=b+\delta b$ with $Ax=b$ and $\hat A \hat x=\hat b \ $.

Here $||.||$ denote the both vector norm and induced matrix norm.

Show that $ \large \frac{||\delta x||}{||\large \hat x||} \leq \kappa (A) \left(\frac{||\delta A||}{||A||}+\frac{||\delta b||}{||\hat b||}+\frac{||\delta A|| || \large \delta b||}{||A|||| \large \hat b||} \right)$, where $\kappa(A)=||A||||A^{-1}||. $

Help me to show it


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.