0

Let be $ \|\cdot \| $ an induced matrix norm and $A, B\in \mathbb{R}^{n,n}$ where that $A$ is invertible. Further more it is $ \|B\| < \|A^{-1}\|^{-1} $. Then $ A+B $ is invertible.

My idea was to show this: If $ (A+B)\cdot x=0 $ then it follows $ x=0\in \mathbb{R}^n $.

So I started with $ 0=\|(A+B)\cdot x\| $ and wanted to show that $ x=0 $ but I don't how I can use this estimation which mentioned above.

hallo007
  • 545

1 Answers1

1

$(I+A^{-1}B)x=0$, $x \neq 0$ implies $\|x\|=\|A^{-1}Bx\|\leq \|A^{-1}\||B||\|X|| <\|x\|$ a contradiction. Hence $I+A^{-1}B$ is invertible. This implies $A+B=A(I+A^{-1}B)$ is invertible.

Kavi Rama Murthy
  • 311,013