I almost every book about quadratic form we can see it described as following function:
$$ f(x) = \frac{1}{2}x^T A x - b^Tx + c $$
My question is:
How can we deduce this formula?
I understand, that the quadratic form is the following product:
$$ Q( \boldsymbol{x} ) = \boldsymbol{x}^TA \boldsymbol{x} $$
where $\boldsymbol{x}$ is a vector, $A$ is a symmetric matrix and $\boldsymbol{x},A$ are defined over $F$, where $F$ is some arbitrary field (except those, who have a characteristic $2$).
This product can be rewritten by summation like this:
$$ Q \left(\boldsymbol{x} = (x_1, \ldots, x_n) \right) = \sum^n_{i=1} \sum^n_{j=1} A_{ij}x_i x_j $$
I understand, that because the matrix is symmetric $A_{ij} = A_{ji}$, so when I want to get the value for one $x$, I must divide it by $2$. That's exactly the first part of the polynomial:
$$ \frac{1}{2}x^T A x $$
But, where do $-b^T + c$ come from? I think, that it's related with the linear system $A \boldsymbol{x} = \boldsymbol{b}$, but I have no idea, how to use it in this deduction...
Suppose we have two variables $x_1$ and $x_2$. Then the general quadratic function is:
$f(x_1,x_2)=a_1x_1^2+a_2x_2^2+a_{12}x_1x_2+b_1x_1+b_2x_2+c$
The first three summnads can be expressed by $\frac12 x^TAx$ with $A=\begin{pmatrix}{} 2a_1 &a_{12} \\ a_{12} & 2a_2\end{pmatrix}$
The fourth and the fifth summand are $b^Tx=\begin{pmatrix}{}b_1 & b_2\end{pmatrix}\cdot \begin{pmatrix}{} x_1 \\ x_2\end{pmatrix}$
And $c$ is a scalar (constant).
Calculate $\frac12x^TAx+b^Tx+c$ and you´ll get $f(x_1,x_2)$.