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I study about random projection and i m really confuse about the relationship between Frobenius norm and singular values. The book say that the $||M||_f^2 $ and $\sigma$ had a correlation.

I found this

How do you express the Frobenius norm of a Matrix as the squared norm of its singular values?

but i really dont take the meaning. For me if

$||M||_f= \sqrt \sum _i σ_i^2$

the square is just

$||M||_f^2 = \sum _i σ_i^2$

My book say that the relationship is between all singular value $\sigma_1 (M), \sigma_2(M)... $

Someone can help me on this?

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