I am taking an optimization course, but I am not very good with math.
In an example this function is given and gradient and hessian is asked. $$f(\vec{x})=\vec{x}^T\begin{bmatrix} 1 & 2 \\ 4 & 7 \end{bmatrix}\vec{x}+ \vec{x}^T\begin{bmatrix} 3 \\ 5 \end{bmatrix} + 6$$
In the solution, function is rewrited as this directly; $$f(\vec{x})=\frac{1}{2}\vec{x}^T\begin{bmatrix} 2 & 6 \\ 6 & 14 \end{bmatrix}\vec{x}+ \vec{x}^T\begin{bmatrix} 3 \\ 5 \end{bmatrix} + 6$$ Is this some kind of general form? What operations did they used to obtain second matrix?
I encountered way many of these $\vec{x}^TA\vec{x}$ formed expressions and many mathematical operations made with them.(derivate etc.) What are these expressions are called? I want to learn more about these things.
Thanks in advance. Sorry if my question is stupid or too basic.
Quadratic models are typically expressed using the Hessian, which means it should have a symmetric matrix and there should be a factor $1/2$ in front of the quadratic term. To symmetrize $x^TAx$, replace $A$ with $(A+A')/2$ by noting that $x^TAx = (x^TAx + x^TA^Tx)/2 = \frac{1}{2}x^T(A+A^T)x$.