Results 1 to 2 of 2

Thread: Diagonalizing Symmetric Bilinear Forms

  1. #1
    Super Member Deadstar's Avatar
    Joined
    Oct 2007
    Posts
    722

    Diagonalizing Symmetric Bilinear Forms

    Find a 2x2 orthogonal matrix P such that P^{-1} S P is diagonal where

    S=\left(\begin{array}{cc}7&-6\\-6&-2\end{array}\right)

    So here's my working...

    The eigenvalues of S are 10 and -5.
    Thus the eigenvectors are \left(\begin{array}{cc}2\\-1\end{array}\right) and \left(\begin{array}{cc}1\\2\end{array}\right).

    Now i know these are supposed to be scaled by \frac{1}{\sqrt{|\lambda|}} which would surely give me,
    P=\left(\begin{array}{cc}\frac{2}{\sqrt{10}}&\frac  {1}{\sqrt{5}}\\\frac{-1}{\sqrt{10}}&\frac{2}{\sqrt{5}}\end{array}\right)

    But the answer has it as all of them are divided by \sqrt{5}... Why is that? Do you scale all of the entries by the same eigenvalues and just choose any eigenvalue to use?

    Also, am i right in saying S is a type (1,1) matrix?
    And is this because there is a +ve and a -ve eigenvalue, or is it because you compute the determinant of the 1x1 matrix (i.e, the number 7), and the determinant of the 2x2 matrix (i.e, S) and since one is +ve and one is -ve that means its type (1,1)?
    Last edited by Deadstar; Apr 8th 2009 at 09:14 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    The scaling factor has nothing to do with the eigenvalue. It is the length of the eigenvector, which for both of these eigenvectors is \sqrt{2^2+1^2} = \sqrt5.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Bilinear Forms and Linear Transformations
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Aug 14th 2010, 11:38 AM
  2. basic bilinear forms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 4th 2010, 03:40 AM
  3. bilinear forms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 12th 2010, 07:58 AM
  4. Bilinear Forms
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Jan 12th 2010, 12:15 AM
  5. Bilinear Forms
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Oct 5th 2009, 02:22 AM

Search Tags


/mathhelpforum @mathhelpforum