Exercise 1848Linear algebra › Quadratic formsL1Added 3 Nov 2026

Diagonalization by congruence

#congruence#diagonalization#quadratic form

Statement

Diagonalize the quadratic form $Q(x,y)=x^2+4xy+3y^2$ by congruence (find an invertible $P$ such that $P^TAP$ is diagonal).

Hint

Use the substitution $x=u-2v$ , $y=v$ that comes from completing the square.

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Solution

Reduction. $Q=(x+2y)^2-y^2$ . Set $u=x+2y$ , $v=y$ . Then $Q=u^2-v^2$ .

Transition matrix. From $u=x+2y$ , $v=y$ : $x=u-2v$ , $y=v$ . So
$P=\begin{pmatrix}1&-2\\0&1\end{pmatrix}.$

Congruence. $A=\begin{pmatrix}1&2\\2&3\end{pmatrix}$ , and
$P^TAP=\begin{pmatrix}1&0\\-2&1\end{pmatrix}\begin{pmatrix}1&2\\2&3\end{pmatrix}\begin{pmatrix}1&-2\\0&1\end{pmatrix}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$

Conclusion. $Q=u^2-v^2$ , signature $(1,1,0)$ .

Diagonalization by congruence

Statement

Solution

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