minimize norm $||C-A^TB ||^2$ by taking derivative with respect to vector

Question

Let's try to minimimize a matrix function $f(B)=||C-A^TB ||$ with respect to $B$. Here $C_{k \times 1}$ and $A_{3 \times k}$ and $B_{3 \times 1}$ matrices. One solution presented to me was to take the matrix norm squared: $$||C-A^TB ||^2= C^T C + A^TBB^{-1}A - 2A^{T}BC$$ and then differentiate by $B$. That's the argument that was presented to me. $$ 2AA^T B - 2AC =0 \text{ so that } B=(AA^T)^{-1}AC $$ How can I make rigorous? Here are some possibilities:

• matrix derivative
• directional derivative
• Fréchet derivative
• Gâteaux derivative