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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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2 Answers 2

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 $1$.

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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.

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