I've been playing around with Symmetric matrices and orthogonal bases of said Symmetric matrices, but I cannot figure out how to find the coordinates.
So, let's say that I have a quadratic function : $6x^{2}_1 + 4x_1x_2+3x^2_2$.
Well now I know that this forms a matrix of $~~~A=~\begin{pmatrix}6 & 2 \\ 2 & 3\end{pmatrix}$.
So now I have to find the eigenvalues of this matrix which are $\lambda_1 = 7 , \lambda_2 = 2$ .
and now I have two Eigenbases of $E_7 = \mathrm{span}\left(\begin{pmatrix}-2\\-1\end{pmatrix}\right), E_2 = \mathrm{span}\left(\begin{pmatrix}~~~1\\-2\end{pmatrix}\right)$.
and the orthogonal bases of these two are just the lengths multiplied by the matrices, so
$$ \vec{w_1} = \frac{1}{\sqrt5}\begin{pmatrix}-2\\-1\end{pmatrix} \qquad \qquad \vec{w_2} = \frac{1}{\sqrt5}\begin{pmatrix}~~~1\\-2\end{pmatrix} $$
But the problem I'm having is that I need an equation $q(\vec{w1} ~c_1 +\vec{w2}~c_2) = c_1^2+c_2^2$.
But how do I find these coordinates? I read my book and it wasn't clear on what I'm supposed to be doing to obtain them.
The quadratic form is written
$$v^TMv$$ and after diagonalization
$$v^TP\Lambda P^Tv$$ or
$$(P^Tv)^T\Lambda (P^Tv).$$
So with
$$w:=P^Tv$$ the form reduces to
$$w^T\Lambda w.$$
Finally
$$\lambda_1w_1^2+\lambda_2w_2^2=(\sqrt{\lambda_1}w_1)^2+(\sqrt{\lambda_2}w_2)^2=c_1^2+c_2^2.$$