Quadratic form

Definition

Suppose that,

$$A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& k\end{array}\right]$$

Then $X^{T}AX$ resolves into

$$q_A(x,y,z)=a{x}^2+e{y}^2+k{z}^2+(b+d)xy+(c+g)xz+(f+h)yz$$

If $A$ is a symmetric matrix then the equation becomes:

$$A=\left[\begin{array}{ccc}a& \frac{b+d}{2}& \frac{c+g}{2}\\ \frac{b+d}{2}& e& \frac{f+h}{2}\\ \frac{c+g}{2}& \frac{f+h}{2}& k\end{array}\right]$$