How to express a quadric equation from canonical form to a different basis.

Question

I have the quadric $3X^2-Y^2-Z^2=0$ expressed in the canonical form, and the matrix of change of basis from a basis B to the canonical form is $$P=\begin{bmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} &0\\ 0 & 0 &-1\end{bmatrix}$$ Now, I would like to express the quadric in respect to the basis B. How do I do that?

Answer 1

Notation: I will use the subscript c meaning that the object is considered in the canonical basis, while b if considered in basis b, and $x_c=(X,Y,Z)$.

The quadratic form is indipendent from basis, so we can write

$x_c^tA_cx_c=x_b^tA_bx_b$.

Using the relation $Px_c=x_b$, we get:

$(Px_c)^tA_b(Px_c)=x_c^t \left(P^t A_b P\right)x_c=x_c^tA_cx_c$

Thus, $A_b=\left(P^{-1}\right)^tA_cP^{-1}$

Note: while in this case your change of basis matrix was symmetric and orthogonal,so there was no difference in $P, P^{-1}, P^t$, you have in general to pay close attention to this differences. Those difficulties lead to the concept of contravariant and covariant transformation