Representation of a two-variable quadratic function

Symmetric Matrices > Definitions > Example

The quadratic function $q : mathbf{R}^2 rightarrow mathbf{R}$ , with values

q_1 (x) = 4x_1^2 + 2x_2^2 + 3 x_1x_2 + 4 x_1 + 5 x_2 + 2 ,

can be represented via a symmetric matrix, as

$q(x) = left(begin{array}{c} x_1 x_2 1 end{array}right)^T left(begin{array}{ccc} 4 & 3/2 & 2 3/2 & 2 & 5/2 2 & 5/2 & 2 end{array}right) left(begin{array}{c} x_1 x_2 1 end{array}right).$

In short:

$q(x) = left( begin{array}{c} x hline 1 end{array} right)^T left( begin{array}{c|c} A & b hline b^T & c end{array} right)left( begin{array}{c} x 1 end{array} right) ,$

where x=(x_1,x_2) , and

$A = left( begin{array}{c|c} 4 & 3/2 hline 3/2 & 2 end{array} right) = A^T, ;; b = left( begin{array}{c} 2 hline 5/2 end{array}right), ;; c = 2.$