Given the real vector $\mathbf{b}$ of dimension $N$ and the real non-symmetric and non-singular $(N,N)$ matrix $\mathbf{H}$, the following function of $\mathbf{w}$ (real vector of dimension $N$) is reported below. \begin{equation} f(\mathbf{w})=\mathbf{H}\mathbf{w}-\mathbf{b} \end{equation} I am now interested in finding the primitive $F(\mathbf{w})$ of $f(\mathbf{w})$ by solving the integral below.
\begin{equation} F(\mathbf{w})=\int(\mathbf{H}\mathbf{w}-\mathbf{b})^\dagger d\mathbf{w} \end{equation}
If $\mathbf{H}$ was symmetric I would say that the primitive would be
\begin{equation} F(\mathbf{w})=\frac{1}{2}\mathbf{w}^\dagger\mathbf{Hw}-\mathbf{b}^\dagger\mathbf{w}+const \end{equation} but the non-symmetry of $\mathbf{H}$ makes me very confused. What are the general rules that need to be applied in the above integration? Any help is appreciated!
Functions like $Hw$ arising from non-symmetric matrices do not have an anti-derivative. For suppose $\nabla F = Hw$, then $$\frac{\partial^2F}{\partial w_i\partial w_j}=\frac{\partial}{\partial w_i}(Hw)_j=H_{ij},\qquad\frac{\partial^2F}{\partial w_j\partial w_i}=\frac{\partial}{\partial w_j}(Hw)_i=H_{ji}$$
To take a specific example, $f(x,y)=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}y\\-x\end{pmatrix}$. Then if $\nabla F=f$, $$\partial_x\partial_yF=\partial_x(-x)=-1\ne1=\partial_y(y)=\partial_y\partial_xF$$