Results 1 to 5 of 5

Thread: matrix forms of quadratic equations

  1. #1
    Member
    Joined
    May 2008
    Posts
    77

    matrix forms of quadratic equations

    I have a problem with determining eigenvalues. This is what I've got thus far:

    Identify and sketch the graph of the quadratic equation
    4x + 10xy + 4y = 9

    This gives the matrix form:
    $\displaystyle \begin{pmatrix} 4 & 5 \\
    5 & 4 \\
    \end{pmatrix}$
    Now we find the eigenvalues:
    Det(A – xI) = $\displaystyle \begin{pmatrix} (4-x) & 5 \\
    5 & (4-x) \\
    \end{pmatrix}$
    = x – 8x – 9
    = (x – 9)(x + 1)
    eigenvalues are $\displaystyle \lambda1 = 9 and + \lambda1 = -1$

    From there, it's pretty simple solving:
    $\displaystyle \lambda1x'^2 + \lambda2y'^2 = 9$

    My problem here is: How do I know which eigenvalue is which? It obviously makes quite a bit of difference to the final result. Nothing in my textbook says.
    Last edited by Dr Zoidburg; May 17th 2008 at 10:17 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by Dr Zoidburg View Post
    I have a problem with determining eigenvalues. This is what I've got thus far:

    Identify and sketch the graph of the quadratic equation
    4x + 10xy + 4y = 9

    [This gives the matrix form:
    $\displaystyle \begin{pmatrix} 4 & 5 \\
    5 & 4 \\
    \end{pmatrix}$
    Now we find the eigenvalues:
    Det(A xI[font=Arial]) = $\displaystyle \begin{pmatrix} (4-x) & 5 \\
    5 & (4-x) \\
    \end{pmatrix}$
    = x 8x 9
    = (x 9)(x + 1)
    eigenvalues are $\displaystyle \lambda1 = 9 and + \lambda1 = -1$

    From there, it's pretty simple solving:
    $\displaystyle \lambda1x'^2 + \lambda2y'^2 = 9$

    My problem here is: How do I know which eigenvalue is which? It obviously makes quite a bit of difference to the final result. Nothing in my textbook says.
    Hmm.. I think that depends on how you wrote the matrix form in the first step
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2008
    Posts
    23
    Quote Originally Posted by Dr Zoidburg View Post
    I have a problem with determining eigenvalues. This is what I've got thus far:

    Identify and sketch the graph of the quadratic equation
    4x + 10xy + 4y = 9

    This gives the matrix form:
    $\displaystyle \begin{pmatrix} 4 & 5 \\
    5 & 4 \\
    \end{pmatrix}$
    Now we find the eigenvalues:
    Det(A – xI) = $\displaystyle \begin{pmatrix} (4-x) & 5 \\
    5 & (4-x) \\
    \end{pmatrix}$
    = x – 8x – 9
    = (x – 9)(x + 1)
    eigenvalues are $\displaystyle \lambda1 = 9 and + \lambda1 = -1$

    From there, it's pretty simple solving:
    $\displaystyle \lambda1x'^2 + \lambda2y'^2 = 9$

    My problem here is: How do I know which eigenvalue is which? It obviously makes quite a bit of difference to the final result. Nothing in my textbook says.
    The key point is tracing back the transformation that you have made by diagonalizing the symmetric matrix $\displaystyle A$.
    Since $\displaystyle \lambda_1=9,\lambda=-1$ are eigenvalues, we find their corresponding eigenvectors. If we do so, we can choose $\displaystyle v_1=\left [\begin{array}{c}1\\ 1\end{array}\right ], v_2=\left [\begin{array}{c}1\\ -1\end{array}\right ]$ where the indices correspond. Now form the matrix $\displaystyle P$ using $\displaystyle v_1,v_2$, i.e., $\displaystyle P=\left [\begin{array}{cc}1&1\\ 1&-1\end{array}\right ]$. Let $\displaystyle D=\left [\begin{array}{cc}9&0\\ 0&-1\end{array}\right ]$, so $\displaystyle PAP^{-1}=D$. Any solution $\displaystyle w=\left [\begin{array}{cc}x&y \end{array}\right ]$ satisfies the equation $\displaystyle wAw^t=9$. Note that $\displaystyle P^{-1}=1/2P^t$, so $\displaystyle wAw^t=wP^{-1}DPw^t=1/2wP^tDPw^t=1/2(wP^t)D(wP^t)^t=9$. So solutions $\displaystyle w$ correspond to solutions of $\displaystyle zDz^t=18$, i.e., the equation $\displaystyle 9{z_1}^2-{z_2}^2=18$, via the transformation $\displaystyle wP^t=z$.

    Hope this helps.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    May 2008
    Posts
    23
    To answer your question, the choice of $\displaystyle \lambda_1,\lambda_2$ is not "crucial", because it will be taken care by the transformation matrix $\displaystyle P$.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    May 2008
    Posts
    77
    Quote Originally Posted by Isomorphism View Post
    Hmm.. I think that depends on how you wrote the matrix form in the first step
    I got it from the quadratic equation:$\displaystyle \begin{pmatrix} a & b \\b & c \\\end{pmatrix}$
    where a,b,c,d are
    $\displaystyle ax^2 + 2bxy + cy^2 = d$

    I should explain more:
    once the equation
    $\displaystyle \lambda1x'^2 + \lambda2y'^2 = d$
    has been reached, I need to state what graphic function it is and plot the graph. So it would appear that knowing what $\displaystyle \lambda1$ and $\displaystyle \lambda2$ are important as I get either
    $\displaystyle 9x'^2 - y'^2 = 9$ giving $\displaystyle x'^2 - y'^2/9 = 1$
    or
    $\displaystyle -x'^2 + 9y'^2 = 9$ giving $\displaystyle -x'^2/9 + y'^2 = 1$
    which lead to vastly different graphs. As I said, there's nothing mentioned in my lecture notes about this, and because I'm doing this paper by correspondence I can't go speak to the lecturer.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. quadratic equations in matrix form.
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Feb 12th 2011, 03:38 AM
  2. Quadratic Forms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 16th 2010, 03:30 AM
  3. quadratic forms
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: May 4th 2010, 12:48 PM
  4. Quadratic Forms???
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Mar 3rd 2009, 08:35 AM
  5. Quadratic Equations and Forms
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: Dec 12th 2006, 11:16 AM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum