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I have the equation: $Q(x,y) = 5x^2 - 6xy + 5y^2$. The first question is to write this in $ Q(z) = z^{T}Az$, in Matrixnotation.

The matrix I calculated is: $$\begin{bmatrix} 5 & -3 \\ -3 & 5 \end{bmatrix} $$

Then I have to diagonalize it, such that $A = UDU^{T}$ in order to use another Basis (from $z \to \overline{z}$).

[EDIT] I solved this part using the diagonalization, and obtaining $$\begin{bmatrix} 2 & 0 \\ 0 & 8 \end{bmatrix} $$ as the matrix for the new quadric. Now how can I solve the next question?

At the end I have to draw the new curve given by $z^{T}Az = 8$.

I tried to read some documentation but I don't understand how to procede, can someone help? Thanks!

Ergo
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$$\begin{pmatrix}5&-3\\-3&\;\;5\end{pmatrix}\stackrel{R_2+\frac35R_1}\rightarrow\begin{pmatrix}5&-3\\0&\frac{16}5\end{pmatrix}\stackrel{C_2+\frac35C_1}\rightarrow\begin{pmatrix}5&0\\0&\frac{16}5\end{pmatrix}$$

and there you are...

DonAntonio
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  • Don, you are one of the few who seems familiar with this method, congruence instead of similarity. What would you call it, and where did you see it? Please compare my http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr – Will Jagy Aug 02 '16 at 17:50
  • @WillJagy I'm not sure about names, yet the basic fact is that multiplication from the left affects the rows, whereas multiplication from the right affects the columns. Thus, carrying on reductions on rows and then exactly the same on columns is, for a symmetric matrix, exactly the way to diagonalize it (which is always possible, say by Jacobi or Sylvester theorems), and getting a congruence relation instead, as you say, similarity. I think this is more or less standard stuff in bilinear forms/quadratic forms in a little advanced linear algebra. – DonAntonio Aug 02 '16 at 22:17
  • Thanks. I've seen many students on this site fumble with this type of problem; I mostly do quadratic forms over the integers, so finding eigenvalues and eigenvectors is not something I generally bother with; meanwhile it seems that is all many students know. – Will Jagy Aug 02 '16 at 22:26
  • Huh. I followed the link about the duplicate, it seems Lagrange made the method I generally name Hermite. Good to know. – Will Jagy Aug 02 '16 at 22:29
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    @WillJagy I've heard Lagrange's method. I know it since secondary school as "completing the square"...because that's precisely what it is. – DonAntonio Aug 02 '16 at 22:31
  • Yes, there is really not much to it; this method differs only in producing the inverse matrix, completing the square gives $P^T D P = H,$ while the method in your answer most directly tells us $Q$ in $Q^T H Q = D; $ if the two $D$ are the same then $PQ= I.$ Nothing you didn't know. – Will Jagy Aug 02 '16 at 22:38