Matrix divided by its Euclidean norm $\frac{x}{\|x\|_{2}}$ I know that dividing a vector by its Euclidean norm
$$\frac{x}{\|x\|_{2}}$$
gives a unit vector. Is there any kind of similar logic that can be applied to matrices? For example:
$$\frac{A^{T}A}{\|A\|^{2}}$$

Matrices of a given shape, not even necessarily square, already comprise a vector space you may as well treat like any other. Its usual definition of a squared Euclidean length is $\sum_{ij}A_{ij}^2$, like the usual $\sum_kv_k^2$, by considering $ij$ to be $k$. This is of course $\sum_{ij}A_{ij}A^T_{ji}=\sum_i(AA^T)_{ii}=\operatorname{tr}AA^T$, or $\operatorname{tr}A^TA$ if you prefer. So the "unit" matrix is just $\frac{A}{\Vert A\Vert},\,\Vert A\Vert:=\sqrt{\operatorname{tr}A^TA}$.
