# Prove the inequality for Condition number of matrix

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 $$\kappa(A)=||A||||A^{-1}||.$$

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