Exercise 1853Linear algebra › Quadratic formsL1Added 3 Nov 2026

Positive definiteness of a 2×2 parameter family

#parameter#positive definite#minors

Statement

For which $m\in\mathbb{R}$ is the quadratic form
$Q(x,y)=x^2+2mxy+4y^2$
positive definite?

Hint

The matrix is $\begin{pmatrix}1&m\\m&4\end{pmatrix}$ . Use Sylvester's criterion.

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Matrix. $A=\begin{pmatrix}1&m\\m&4\end{pmatrix}$ .

Leading principal minors.
- $\Delta_1=1>0$ .
- $\Delta_2=4-m^2>0\iff|m|<2$ .

Conclusion. $Q$ is positive definite iff $|m|<2$ , i.e. $-2<m<2$ .