$Q\left(x_1\, ..\. \, x_n\right) \= {\overrightarrow{x}}^T\cdot A\cdot \overrightarrow{x}$

        $\= \left[x_1 \dots x_n\right]\cdot \left[\begin{array}{ccc}{a}_{11}& \dots & a_{1 n}\\ & ⋮& \\ a_{n 1}& \dots & a_{n n}\end{array}\right]\cdot \left[\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right]$

                   $\= \left[x_1 \dots x_n\right]\cdot \left[\begin{array}{c}\sum a_{1 i}\mathbf{}\cdot x_i\\ ⋮\\ \sum _{}{a}_{n i}\mathbf{}\cdot x_i\end{array}\right]$

        $\= a_{11}\cdot x_1^2 \+ a_{12}\cdot x_1\cdot x_2 \+ ..\. \+ a_{1 n}\cdot x_1\cdot x_n \+ a_{21}\cdot x_2\cdot x_1 \+ a_{22}\cdot x_2^2 \+ ..\. \+ a_{2 n}\cdot x_2\cdot x_n \+ ..\. ..\. ..\. \+ a_{\mathrm{n1}}\cdot x_n\cdot x_1 \+ a_{n 2}\cdot x_n\cdot x_2 \+ a_{n n}\cdot x_n^2$

        $\=\sum _{i\,j\=1}^n{a}_{i j}\cdot x_i\cdot x_j$

If $\overrightarrow{\mathit{x}}$ has only two elements, $\overrightarrow{\mathit{x}}\mathbf{ }\mathbf{\=}\mathbf{ }\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right] \=\left[\begin{array}{c}x\\ y\end{array}\right]$, then we can graphically represent the quadratic form, $Q\left(x\,y\right)$, as a function ${\mathrm{ℝ}}^2 \to \mathrm{ℝ}$.  This is shown in the plot below.

This also allows us to visually determine the classification of the $2\times 2$ symmetric matrix A as:

Using quadratic forms to classify matrices as definite, semi-definite, or indefinite can be useful in performing the multivariable second derivative test.

Let $f\: {\mathrm{\ℝ}}^2 \to \mathrm{\ℝ}$ have continuous second partial derivatives in some neighborhood of a critical point $\left(a\,b\right)$ and let $H\left(a\,b\right) \= \left[\begin{array}{c}{f}_{\mathrm{xx}}\(a\,b\) f_{\mathrm{xy}}\(a\,b\)\\ f_{\mathrm{yx}}\(a\,b\) f_{\mathrm{yy}}\(a\,b\)\end{array}\right]$ be the Hessian matrix of $f$ evaluated at $\left(a\,b\right)$.