Consider the symmetric bilinear form $f(x,y)=x_1y_1+5x_2y_2+x_3y_3+2x_1y_2+2x_2y_1+3x_1y_3+3x_3y_1$.I am asked to find an orthogonal basis for this symmetric bilinear form.Hence,I am asked to find a matrix $P$ such that $P^tAP$ is diagonal,where $A=\begin{pmatrix}1 &2&3\\2& 5&0\\3&0&1\end{pmatrix}$.I have done this problem and I want to know whether it is correct or not.
$f(x,x)=x_1^2+5x_2^2+x_3^2+4x_1x_2+6x_1x_3$ .We observe that for $x=(1,0,0)$ ,$f(x,x)=1\neq 0$.So we take $\alpha_1=(1,0,0)$,Now we calculate the orthogonal complement of $W=\{\alpha_1\}$ which is $\{(x_1,x_2,x_3)\in \mathbb R^3: x_1+2x_2+3x_3=0\}$.We take a vector from this orthogonal complement viz. $\alpha_2=(2,-1,0)$.We now find vector orthogonal to both $\alpha_1$ and $\alpha_2$ which is a solution of $x_1+2x_2+3x_3=0$ and $-x_2+6x_3=0$.We find a vector $(-15,6,1)$ satisfying the $2$ equations.Take it as $\alpha_3$.Now by constrution,$\{ \alpha_1,\alpha_2,\alpha_3\}$ is an orthogonal basis.Now the matrix $P$ that would make $A$ into diagonal matrix is $P=\begin{pmatrix}1 & 2 & -15\\0 & -1 & 6\\0 & 0 &1\end{pmatrix}$.Is my solution correct?