Representation of a two-variable quadratic function

L. El Ghaoui

Representation of a two-variable quadratic function

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

$q(x) = 4x_1^2 + 2x_2 ^2 +3x_1 x_2 +4x_1 + 5x_2 + 2,$

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:

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

where , and

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

Representation of a two-variable quadratic function

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