Conic Equation using a Quadratic Form

  • #1

Homework Statement



x12+x1x2+2x22=8

a) Write the equation using a quadratic form i.e. [itex]\underline{x}[/itex]TA[itex]\underline{X}[/itex]=8

b)Find the Matrix Q such that the transformation [itex]\underline{X}[/itex]=Q[itex]\underline{Y}[/itex] diagonalises A and reduces the quadratic form to standard form in terms of coordinates (y1,y2)

Homework Equations



[itex]\underline{X}[/itex]=Q[itex]\underline{Y}[/itex]
[itex]\underline{X}[/itex]TA[itex]\underline{X}[/itex]=8

The Attempt at a Solution



For question b), I got the A matrix as [1 1;0 2] or [1 0.5;0.5 2] *sorry, don't know how to use the matrix operator so I've written it matlab style*.

I used the first matrix to give a better looking eigenvalues, which resulted in 2 and 1. From the values, I got a vector of [1;0] and [1;1]

Using the vectors, I got a Q matrix of [1 0; 1/sqrt(2) 1/sqrt(2)]
and using [itex]\underline{X}[/itex]=Q[itex]\underline{Y}[/itex], I got
2.707y12+2.707y1y2+y22 which I'm not even sure if its right.

Could someone please assist me in tackling this question?
Thanks!
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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Yes, the matrix here is
[tex]\begin{bmatrix}1 & 0.5 \\ 0.5 & 2\end{bmatrix}[/tex]

The first matrix you give is wrong. In order to be certain that there are eigenvalues, you must have a symmetric matrix. You don't want "better looking" eigenvalues, you want the right eigenvalues!

Finally, the problem asked you to "Find the Matrix Q such that the transformation X=QY diagonalises A reduces the quadratic form to standard form in terms of coordinates (y1,y2)" but your final result is NOT in standard form.
 
  • #3
Yes, the matrix here is
[tex]\begin{bmatrix}1 & 0.5 \\ 0.5 & 2\end{bmatrix}[/tex]

The first matrix you give is wrong. In order to be certain that there are eigenvalues, you must have a symmetric matrix. You don't want "better looking" eigenvalues, you want the right eigenvalues!

Finally, the problem asked you to "Find the Matrix Q such that the transformation X=QY diagonalises A reduces the quadratic form to standard form in terms of coordinates (y1,y2)" but your final result is NOT in standard form.

Got it! Thanks!
 

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