Exercise 1149Linear algebra › Quadratic formsL1Added 3 Nov 2026

An indefinite quadratic form

#indefinite#signature#quadratic form

Statement

Show that the quadratic form
$Q(x,y) = x^2 - y^2$
is indefinite (neither positive nor negative).

Hint

Find vectors $u,v$ with $Q(u)>0$ and $Q(v)<0$ .

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Solution

Positive value. $Q(1,0)=1>0$ .

Negative value. $Q(0,1)=-1<0$ .

Conclusion. $Q$ takes both positive and negative values, so it is indefinite. Its signature is $(1,1,0)$ : one positive square and one negative square.

Matrix form. $A=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ , with eigenvalues $1$ and $-1$ , of opposite signs.

An indefinite quadratic form

Statement

Solution

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