Consider the quadratic form q($x_1$,$x_2$,$x_3$) := $x_1^2$+$x_3^2$+$2x_1x_2$ $−4x_1x_3$ +$2x_2x_3$, ($x_1,x_2,x_3$)∈ $ℝ^3$.

Question

Consider the quadratic form q($x_1$,$x_2$,$x_3$) := $x_1^2$+$x_3^2$+$2x_1x_2$ $−4x_1x_3$ +$2x_2x_3$, ($x_1,x_2,x_3$)∈ $ℝ^3$.

(a) Find the symmetric matrix $A$ representing q. For (a) I got
$$A=\begin{pmatrix}1&1&-2\\1&0&1\\-2&1&1\\\end{pmatrix}$$

(b) Find a corresponding orthogonal matrix $P$ of eigenvectors of $A$.

For (b) I started by getting the eigenvalues by doing det($A$-$\lambda$$I_3$)=0.

I got $\lambda$ =1, $\lambda$=-2 and $\lambda$=3.

I then found the eigenvector of each eignevalue by doing RREF. For $\lambda$=1 I got (1,2,1), for $\lambda$=-2 I got(1,-1,1) and for $\lambda$=3 I got (-1,0,1).

Am I correct?

(c) Write down the maximum and minimum values taken by q over the unit vectors in $ℝ^3$.

Is the max simply $\lambda$=3 on unit vector $\frac{1}{\sqrt2}$(-1,0,0). And the min is $\lambda$=-2 on the unit vector $\frac{1}{\sqrt3}$(1,-1,1) ?.