# Matrices whose condition number is $1$

The condition number of the identity matrix $$I$$ always equals $$1$$. Are there any other matrices that have a condition number equal to $$1$$, but are neither the identity matrix nor $$\lambda I$$ (for any scalar $$\lambda$$)?

(because if $$A$$ is a matrix, then $$\mbox{cond}(\lambda A) = \mbox{cond}(A)$$)

The $2$-norm condition number (i.e., the ratio of the largest singular value to the smallest singular value) of a unitary matrix is always equal to $1$.

Yes for example random matrix $X_{N\times N}$ has singular values roughly satisfying in reality $$\sigma_1=\sigma_2=\sigma_2,...,=\sigma_N$$ which means that the condition number is $1$.

It is the same thing with taking the fourier transform of a dirac delta function and result in the frequency domain is flat. Similarly when you take the fourier transform of the white noise which is random then you also get a flat spektrum.

For any orthogonal matrix $$O$$, the condition number is equal to $$1$$.

By definition, $$O$$ satisfies $$O^{-1} = O^T$$ or $$O^T O = I$$.

By definition, $$\Vert O \Vert_2 = \sqrt{ \lambda_{\max}(O^T O)} = \sqrt{\lambda_{\max}(I)} = \sqrt{1} = 1$$

Since $$O^{-1} = O^T$$ is also orthogonal, it is immediate that $$\Vert O^{-1} \Vert_2 = 1$$

Hence, for any orthogonal matrix $$O$$, $$\kappa(O) = \Vert O \Vert_2 \Vert O^{-1} \Vert_2 = 1$$ showing that the orthogonal matrices are always well-conditioned.