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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.
