Exercise 1844Linear algebra › Quadratic formsL1Added 3 Nov 2026

Canonical form of a quadratic form

#canonical form#Gauss reduction#signature

Statement

Put the quadratic form
$Q(x,y,z)=x^2+2y^2+3z^2+2xy-4xz$
into canonical form (sum of squares with coefficients $\pm1$ or $0$ ).

Hint

Complete the square in $x$ first (since $x^2$ appears), then in $y$ , then in $z$ .

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Group terms containing $x$ . $x^2+2xy-4xz=x^2+2x(y-2z)=(x+y-2z)^2-(y-2z)^2$ .

Remaining terms.
$Q=(x+y-2z)^2-(y-2z)^2+2y^2+3z^2=(x+y-2z)^2+y^2+4yz-z^2.$

Square in $y$ . $y^2+4yz=(y+2z)^2-4z^2$ . So
$Q=(x+y-2z)^2+(y+2z)^2-5z^2.$

Canonical form. With $u=x+y-2z$ , $v=y+2z$ , $w=z$ :
$Q=u^2+v^2-5w^2.$

Signature. $(2,1,0)$ : two positive squares, one negative.

Conclusion. $Q=u^2+v^2-5w^2$ (or, after rescaling $w'=\sqrt5 w$ , $Q=u^2+v^2-w'^2$ ).