# When is the condition number of an orthogonal matrix equal to $1$

If the norm is induced by an inner product, then:

$$||A||=\max_{||x||=1}||Ax||=\max_{||x||=1}||x||=1$$

where the second equality holds from the fact that if the norm is induced from an inner product, then $$||Ax||=||x||$$ for any orthogonal matrix $$A$$. Similarly, since $$A^{-1}$$ is also orthogonal we get $$||A^{-1}||=1$$ and thus $$cond(A)=||A||\cdot||A^{-1}|| = 1 \cdot 1=1$$.

Is this true that $$cond(A) = 1$$ for any orthgonal matrix $$A$$ for general norms too?

• No, the Frobenius condition number of the identity is $2$. Dec 19, 2019 at 19:07

• The Frobenius norm is submultiplicative and unitarily invariant. We have $$\|I_2\|=\sqrt{2}$$ and hence $$\|I_2\|\|I_2^{-1}\|=2>1$$.
• Let $$P=\pmatrix{1&1\\ 0&1}$$ and $$A=\pmatrix{0&-1\\ 1&0}$$. Then $$\|M\|=\max_{x\ne0}\frac{\|PMx\|}{\|Px\|}=\sigma_\max(PMP^{-1})$$ is an induced norm and $$\|A\|=\frac{3+\sqrt{5}}{2}>1$$. Since $$A^{-1}=-A$$, we have $$\|A\|\|A^{-1}\|=\|A\|^2>1$$.
However, if $$\|\cdot\|$$ is a unitarily invariant induced norm, then $$\|A\|=\|I\|=\max_{x\ne0}\frac{\|Ix\|}{\|x\|}=1$$ for every real orthogonal matrix $$A$$. Hence the condition number of $$A$$ is $$1$$ in this case.