Exercise 1851Linear algebra › Quadratic formsL1Added 3 Nov 2026

Gram matrix of a quadratic form

#Gram matrix#quadratic form#polar form

Statement

Let $\varphi$ be the symmetric bilinear form with matrix
$A=\begin{pmatrix}2&1\\1&3\end{pmatrix}.$
Compute the quadratic form $Q(u)=\varphi(u,u)$ and check that its matrix is $A$ .

Hint

$Q(u)=u^TAu$ .

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Solution

Computation.
$Q(x,y)=(x,y)\begin{pmatrix}2&1\\1&3\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=2x^2+2xy+3y^2.$

Matrix. The matrix of $Q$ in the canonical basis is $A$ , since the coefficient of $x^2$ is $2$ , of $y^2$ is $3$ , and of $xy$ is $2\cdot1=2$ .

Conclusion. $Q(x,y)=2x^2+2xy+3y^2$ with Gram matrix $A$ .

Gram matrix of a quadratic form

Statement

Solution

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