Condition number properties Assuming $A$ is invertible, I know if $A^{-1} = A^T$, then $cond(A)=1$. Is it true the other way around, as in If $cond(A)=1$, $A^{-1}=A^T$. Also, does $cond(A^{-1})$ always equal $cond(A)$?

Hint: If an $n\times n$ matrix $A$ is invertible, then it's easy to check that all eigenvalues are non-zero. Let's enumerate the eigenvalues of $A$ in the following way: $\lambda_1,\lambda_2,\dots,\lambda_n$, such that $$|\lambda_1|\geq|\lambda_2|\geq\dots\geq|\lambda_n|>0.$$ Then you can see that the condition number of $A$ is in fact $$cond(A)=\frac{|\lambda_1|}{|\lambda_n|}.$$ Next, you are going to use the fact that the eigenvalues of $A^{-1}$ are precisely: $\lambda_1^{-1},\lambda_2^{-1},\dots,\lambda_n^{-1}$. Finally, you can obtain the relation immediately: $$cond(A)=cond(A^{-1}).$$

Now if $cond(A)=1$, this means all eigenvalues of $A^TA$ equals $1$ because $A^TA$ is positive definite. Recall that every symmetric matrix is diagonalizable, thus we actually have $A^TA=I$.