Determine its definiteness of the quadratic form $x^2_1 - 2x^2_2 + 5x^2_3 + 2x_1x_2 - 6x_2x_3$

Question

The equation is the following: $x^2_1$ - 2$x^2_2$ + 5$x^2_3$ + 2$x_1$$x_2$ - 6$x_2$$x_3$

I thought about finding the definiteness through the positivity or negativity of eigenvalues. The first step was to converting the quadratic form into a matrix and I got the following:

$A=\begin{bmatrix} 1 & 1 & 0 \\ 1 & -2 & -3 \\ 0 & -3 & 5 \end{bmatrix}$

Then I decided to find the determinant of [A-λI], where I is identity matrix, and equalise it to 0:

$det\begin{bmatrix} 1-λ & 1 & 0 \\ 1 & -2-λ & -3 \\ 0 & -3 & 5-λ \end{bmatrix}=0$

The furthest simplification I could achieve was : $-λ^3+4λ^2+17λ-24=0$

But it is impossible to find λ without using some software.

Did I make a mistake somewhere? If not, which other method can I use to find the definiteness of the quadratic form?

Your replies are greatly appreciated, thank you very much.

Answer 1

Another Approach:

1. Find out the associated symmetric matrix.
2. Convert it to canonical form under congruence.
3. From the canonical form just by inspection you can find the definiteness.