Exercise 266Linear algebra › Quadratic formsHigh schoolAdded 3 Nov 2026

Definition and matrix of a quadratic form

#quadratic form#symmetric matrix#definition

Statement

Write the quadratic form
$Q(x,y) = x^2 + 4xy + 3y^2$
in the form $Q(X) = X^T A X$ , where $A$ is a symmetric matrix.

Hint

The coefficient of $x_ix_j$ for $i\neq j$ is $2a_{ij}$ in $X^TAX$ .

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Identification.
$Q(x,y) = a_{11}x^2 + 2a_{12}xy + a_{22}y^2.$
So $a_{11}=1$ , $2a_{12}=4\Rightarrow a_{12}=2$ , $a_{22}=3$ .

Matrix.
$A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}.$

Verification. $X^TAX = (x,y)\begin{pmatrix}1&2\\2&3\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = x^2+2xy+2xy+3y^2 = x^2+4xy+3y^2$ .

Conclusion. $Q(x,y)=X^TAX$ with $A=\begin{pmatrix}1&2\\2&3\end{pmatrix}$ .