Exercise 1147Linear algebra › Quadratic formsL1Added 3 Nov 2026

A degenerate quadratic form

#degenerate#signature#quadratic form

Statement

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

Hint

$Q$ is a square of a linear form; find the kernel.

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Reduction. $Q=(x+y)^2$ . Let $u=x+y$ . Then $Q=u^2$ , with one positive square.

Kernel. $Q(x,y,z)=0\iff x+y=0$ . The kernel has dimension $2$ (a plane in $\mathbb{R}^3$ ).

Signature. One positive square, zero negative squares, kernel dimension $2$ . Signature: $(1,0,2)$ .

Conclusion. $Q$ is positive but degenerate (not positive definite). Its rank is $1$ , its kernel has dimension $2$ .