Finding Jacobian of $f(x): A\overrightarrow x \bullet B\overrightarrow x$

Question

If A and B are mxm matrices and $f$ is defined as $f: \Bbb{R}^m \rightarrow \Bbb{R}$ by $f(x): A\overrightarrow x \bullet B\overrightarrow x$ . How would one go about finding the Jacobian matrix $J(f)(\overrightarrow c)$ ? Thank you!

Answer 1

Hint: The gradient (as a column vector) of $\mathbf{x}^T M \mathbf{x}$ (where $M$ is a fixed matrix) is $\color{blue}{(M + M^T)\mathbf{x}}$. Try and prove this yourself, or see this link for hints: Gradient of $x^{T}Ax$). Transpose this to get the Jacobian matrix as a $1\times m$ matrix.

To use this fact for your problem, use the fact that $\mathbf{z}\cdot \mathbf{w} = \mathbf{z}^T \mathbf{w}$ (and a fact about the transpose of a product).