Condition number example

Question

I have been learning about the condition number, i.e $$cond_p(A) = ||A||_p||A^{-1}||_p.$$

I've been considering the $2$ norm, and have been thinking about when the condition number is equal to $1$. Of course, when $A = I$ this holds, but I can't think of any other examples?

Thank you.

Answer 1

When $A$ is an orthogonal matrix, then $A^T = A^{-1}$.

Hence, for an orthogonal matrix $A$, we have $A^T A = I$.

It follows that $\lambda_{\max}(A^T A) = 1$.

Hence, for an orthogonal matrix $A$, $\Vert A \Vert_2 = 1$.

(Every orthogonal matrix is well-conditioned).

In a similar way, we can show that $\Vert A^T \Vert_2 = 1$.

(If $A$ is orthogonal, then $A^T$ is also orthogonal.)

Thus, $\Vert A^{-1} \Vert_2 = 1.$

Hence, for any orthogonal matrix $A$,

$\kappa(A) = \Vert A \Vert_2 \, \Vert A^{-1} \Vert_2 = 1$.

(Orthogonal matrices are well-conditioned matrices.)