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Returning to this example, the Frobenius norm is the square root of the sum of the squares of the elements, and is equal to

\[ \|A\|_F=\sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2}=5.4772. \]

The largest singular value norm is simply $\sigma_1=4$. Thus, the Euclidean norm of the output $A x$ cannot exceed four times that of the input $x$.