Exercise 1847Linear algebra › Quadratic formsL1Added 3 Nov 2026

The cone $x^2+y^2-z^2=0$

#cone#quadratic form#signature

Statement

Describe the zero set of the quadratic form $Q(x,y,z)=x^2+y^2-z^2$ and identify its signature.

Hint

The equation $x^2+y^2=z^2$ describes a cone.

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Signature. $Q$ has two positive squares and one negative square: signature $(2,1,0)$ .

Zero set. $Q=0\iff x^2+y^2=z^2$ . This is the (double) cone of revolution about the $z$ -axis, with apex at the origin.

Geometric interpretation. The signature $(2,1,0)$ indicates that the level set $Q=0$ is a cone (not a single point or the empty set).

Conclusion. $Q$ is indefinite with signature $(2,1,0)$ , and $Q=0$ is a double cone.