$\text{cond}(A)\gt \text{cond}(A+B)$ for $AA^T=I$

Question

Let $A$ a matrix such that $AA^T=I$. Is there a matrix $B$ such that

$$\text{cond}(A)\gt \text{cond}(A+B)$$

If so give numerical exmples for this, otherwise prove that there isn't.

The condition number is: $\text{cond}(A)=\Vert A\Vert \Vert A^{-1}\Vert$

Answer 1

Assuming you're using the spectral norm, the condition number of $A$ is $1$. But there is no matrix with condition number less than $1$, because $$1 = \|I\| = \|M M^{-1}\| \le \|M \| \|M^{-1}\|$$ for all invertible matrices $M$.