Exercise 1855Linear algebra › Quadratic formsL1Added 3 Nov 2026

Signature of a quadratic form in 3 variables

#signature#reduction#quadratic form

Statement

Determine the signature of the quadratic form
$Q(x,y,z) = x^2 + 2xy + 2xz + 2y^2 + 4yz + 3z^2.$

Hint

Complete the square in $x$ first, then in $y$ , then in $z$ .

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Square in $x$ . Consider $x^2+2x(y+z)$ , which is $(x+y+z)^2-(y+z)^2$ . So
$Q = (x+y+z)^2 - (y+z)^2 + 2y^2+4yz+3z^2.$

Simplify.
$-(y^2+2yz+z^2)+2y^2+4yz+3z^2 = y^2+2yz+2z^2.$

Square in $y$ . $y^2+2yz+2z^2=(y+z)^2+z^2$ .

Result.
$Q = (x+y+z)^2 + (y+z)^2 + z^2.$

Signature. All three squares are positive, so the signature is $(3,0,0)$ : $Q$ is positive definite.

Conclusion. Signature $(3,0,0)$ .