# Condition number matrix

For the identity matrix $I$, the condition number of the matrix always equals 1.

My question is: are there any other matrices out there that have a condition number equal to 1, but are not the identity matrix or $\lambda I$ (for any scalar $\lambda$) ?
(because if $A$ is a matrix, then $cond(\lambda A) = cond(A)$)

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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 $1$.