Let $A\mathbf{x}=\mathbf{b}$ be an overdetermined system, with $A$ being an $n \times m $ full-column rank rectangular matrix.
Are these minimization problems equivalent?
$$ 1) \;\underset{\mathbf{x}} {\mathop{\min }}\,{{\left\| A\mathbf{x}-\mathbf{b} \right\|}_{2}}$$
$$ 2)\; \underset{\mathbf{x}} {\mathop{\min }}\,({{\left\| A\mathbf{x}-\mathbf{b} \right\|}_{2}})^2 $$
I'm only familiar with the known OLS result which gives rise to the normal equation $\mathbf{x}_{ls}=(A^tA)^{-1}A^t\mathbf{b}$ obtained from $2)$ by expressing it as: $\; \underset{\mathbf{x}} {\mathop{\min }}\ (A\mathbf{x}-\mathbf{b})^t (A\mathbf{x}-\mathbf{b})$ and by imposing gradient to be zero.
I also think that both $1)$ and $2)$ share the convexity property because any "$p-norm$" is convex and a square of a convex non-negative function is still convex.