Diagonalizing Quadratic Forms. Linear Algebra

Question

I have a question that reads:

Diagonalize the quadratic form $A(x,y) = 3x^2 -12xy + 7y^2$ by completing the square.

What is diagonalization? Is that when I should find the eigenvector matrix, say, $S$, find it's inverse $S^{-1}$, and then multiply it by some matrix in order to obtain a diagonal matrix that is easy to find powers of?

I think that's what it is.

Problem is, I guess I don't understand what it means by "completing the square". I mean I know what completing the square is in, say, high school algebra, but I don't understand what it means in this context.

Any clarification is greatly appreciated.

Thanks!

Answer 1

Diagonalisation is to write it without mixed terms $xy$. $$3x ^2-12xy+7y^2=3(x^2-4xy)+7y^2=3(x-2y)^2-5y^2$$

Answer 2

You need to find a new pair of variables, call them $u$ and $v$, each of which is a linear combination of $x$ and $y$, such that $A(u,v)$ does not contain cross terms (containing the product $uv$). If you write out $A$ as a quadratic form, $$ A(x,y) = \begin{bmatrix} x\\y\end{bmatrix}^T\begin{bmatrix}3 & -4\\-4 & 7 \end{bmatrix}\begin{bmatrix} x\\y\end{bmatrix}$$ If you diagonalize the matrix in the middle, the eigenvectors give you the linear combinations, and the eigenvalues are the new coefficients to $u^2$ and $v^2$.