Condition number of matrix is $1$ iff $A^TA=\alpha I$

Question

I am trying to prove the following:

The condition number $\kappa_p(A)=1$ for $A$ an matrix iff $A^TA=\alpha I$ for some scalar number $\alpha\neq 0$.

I read somewhere on the internet that I somehow have to involve the singular value decomposition but I can't seem to figure out how.

Answer 1

Note that if $\kappa$ is the condition number (with respect to the Euclidean norm), then $$ \kappa(A) = \frac{\sigma_1(A)}{\sigma_n(A)} $$ Where $\sigma_1$ and $\sigma_n$ denote the highest and lowest singular values of $A$, respectively. It's clear that $\kappa(A) = 1$ if and only if $\sigma_1 = \sigma_n$.

Now, consider the singular value decomposition.