Is minimizing the norm of a vector equivalent to minimizing the square of the norm?

Question

Are the following two problems equivalent?

$$ \arg \min_{y} || A \vec{x} - \vec{y} || $$

and

$$ \arg \min_{y} || A \vec{x} - \vec{y} ||^2 $$

I feel like they are equivalent as it is just the square of a scalar ($|| A \vec{x} - \vec{y} || $) but I am never sure how things behave in higher dimensions (i.d. $\vec{y}$).

Answer 1

Yes. Even though $\vec{y}$ may come from a multidimensional space, a norm is always a map from some space $X$ to the nonnegative real numbers, i.e. $\| \cdot \| \colon X \to \mathbb{R}_+$. Therefore they are equivalent.