Let $A\in M_n(\mathbb R)$ be an orthogonal matrix. Then: $cond (A) =1$. I tryed to use facts about the eigenvalues but is did not help. In 2-norm it is easy to prove it since $||A||_2 = \sqrt{\rho (A^T A)}=\sqrt{\rho (I)}=1$. What about a general norm? Thanks for helpers!
$\endgroup$1 Answer
$\begingroup$It is known that $cond(A)= \|A\|\|A^{-1}\|$, so if you want to prove that $cond(A) = 1$, it would be sufficient to prove that $\|A^{-1}\| = \frac{1}{\|A\|}$. Perhaps this path will lead you to a simpler answer.
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