Let $r>0$, and let $A\notin\mathcal{B}_r(0)\triangleq\{Y\in\mathbb{R}^{n\times n}:\|Y\|_2\leq r\}$, where $\|\cdot\|_2$ is the induced 2-norm. Let $\bar A$ be the orthogoanl projection of $A$ on the boundary of $\mathcal{B}_r(0)$ defined by $$\bar A\triangleq{\rm argmin}_{X\in\partial \mathcal{B}_r(0)}\|X- A\|_{\rm F},$$ where $\partial \mathcal{B}_r(0)$ denotes the boundary of $\mathcal{B}_r(0)$, and $\|\cdot\|$ denotes the Frobinius norm.
How can I orthogonally project $A$ to the boundary of $\mathcal{B}_r(0)$?
I know that if, in the definition of $\mathcal{B}_r(0)$, we replace $\|\cdot\|_2$ by $\|\cdot\|_{\rm F}$ , then $\bar A=\dfrac{Ar}{\|A\|_{\rm F}}$ is the orthogoanl projection of $A$ on the boundary of $\mathcal{B}_r(0)$. Can we show that $\bar A=\dfrac{Ar}{\|A\|_{2}}$ also works here?