matrix differentiation 2

Question

Define matrix $\mathbf{A} \in \mathbb{C}^{n \times k}$, matrix $\mathbf{B} \in \mathbb{C}^{m \times k}$, and vector $\mathbf{c} \in \mathbb{C}^{k \times 1}$. Also, assume that $\mathbf{A}$ and $\mathbf{c}$ are independent of $\mathbf{B}$. What is the derivative of $\mathbf{A}\mathbf{B}^T\mathbf{B}\mathbf{c}$ respect to $\mathbf{B}$.

Answer 1

$\def\C{\mathbf C}\def\M#1#2{\operatorname{Mat}_{#1,#2}(\C)}$Denote by $f \colon \M mk \to \C^n$ the function in question, that is $$ f(B) = AB^tBc $$ We have for any $B, H \in \M mk$ that \begin{align*} f(B+H) - f(B) &= A(B^t + H^t)(B + H)c - AB^tBc\\ &= AH^tBc + AB^tHc + AH^tHc \end{align*} Now, note that $\def\norm#1{\left\|#1\right\|}$by submultiplikativity and continuity of $B \mapsto B^t$, we have $$ \norm{AH^tHc} \le \norm A \norm H^2 \norm c = o(\norm H), \qquad H \to 0$$ That is, the (Frechet) derivative of $f$ at $B$ is the function $$ f'(B) \colon \M mk \to \C^n, \qquad H \mapsto AH^t Bc + AB^tHc $$