Frobenius norm and singular values

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?