Form definiteness with a parameter
Task number: 4453
Given a real quadratic form
\(g((x_1, x_2, x_3)^T) = x_1^2 + 2x_1x_2 + 2x_1x_3 + 2x_2^2 + ax_2x_3 + 5x_3^2\).
For which values of the parameter \(a\in \mathbb R\) is this form positive definite (i.e. \(\forall \boldsymbol u\in\mathbb R^3\setminus\boldsymbol 0: g(\boldsymbol u)\gt 0\)) and for which
values is it negative definite
(i.e. \(\forall \boldsymbol u\in\mathbb R^3\setminus\boldsymbol 0: g(\boldsymbol u)\lt 0\))?
Solution
We construct the form matrix and modify it by Gaussian elimination:
\( \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & \frac{a}2 \ 1 & \frac{a}2 & 5 \end{pmatrix} \sim\sim \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & \frac{a}2-1 \ 0 & \frac{a}2-1 & 4 \end{pmatrix} \sim\sim\sim \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -\frac{a^2}4+a+3 \end{pmatrix} \)The form is positive definite if it has the signature \((3{,}0,0)\), that is, if \(-\frac{a^2}4+a+3\gt 0\).
It cannot be negatively definite because the first component of the signature is at least 2, and you would need to get the signature \((0{,}3,0)\).
Answer
The form is positive definite for \(a\in(-2{,}6)\). It is not negative definite for any choice of \(a\).
