Exercise 2214Linear algebra › Quadratic formsL1Added 3 Nov 2026

Signature depending on a parameter

#signature#parameter#quadratic form

Statement

For which $a\in\mathbb{R}$ is the quadratic form
$Q(x,y,z)=x^2+y^2+z^2+2a(xy+yz)$
positive definite? Find the signature for all $a$ .

Hint

Reduce by grouping $x^2+2axy+...$ , completing the square in $x$ first.

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Solution

Group in $x$ . $x^2+2axy=(x+ay)^2-a^2y^2$ . So
$Q=(x+ay)^2+(1-a^2)y^2+z^2+2ayz.$

Group in $z$ . $z^2+2ayz=(z+ay)^2-a^2y^2$ . So
$Q=(x+ay)^2+(1-2a^2)y^2+(z+ay)^2.$

Signature analysis.
- If $1-2a^2>0$ , i.e. $|a|<1/\sqrt2$ : signature $(3,0,0)$ , positive definite.
- If $a=\pm1/\sqrt2$ : signature $(2,0,1)$ , degenerate positive.
- If $|a|>1/\sqrt2$ : signature $(2,1,0)$ , indefinite.

Conclusion. $Q$ is positive definite iff $|a|<1/\sqrt2$ .

Signature depending on a parameter

Statement

Solution

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