Results 1 to 2 of 2

Math Help - finding polar decomposition!

  1. #1
    Junior Member
    Joined
    Feb 2009
    Posts
    40

    finding polar decomposition!

    I need help finding the polar
    decomposition
    of

     <br /> <br />
\left( \begin{array}{cc} 11 & -5 \\ -2 & 10\end{array} \right)<br />
    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
    7
    If A = \begin{bmatrix}11 & -5 \\ -2 & 10\end{bmatrix} then the polar decomposition of A is a factorisation A = UR, where U is unitary and R is positive. To find R use the fact that R^2 = A^*A = \begin{bmatrix}125 & -75 \\ -75 & 125\end{bmatrix}. So R is the positive square root of that matrix, which you can compute by diagonalising it. You should find that the eigenvalues of R^2 are 50 and 200, with corresponding normalised eigenvectors \frac1{\sqrt2}\begin{bmatrix}1 \\ 1\end{bmatrix} and \frac1{\sqrt2}\begin{bmatrix}1 \\ -1\end{bmatrix}. So R^2 = PDP^{-1}, where P = P^{-1} = \frac1{\sqrt2}\begin{bmatrix}1 &1  \\ 1 & -1\end{bmatrix} and D = \begin{bmatrix}50 & 0 \\ 0 & 200\end{bmatrix}. The square root is given by R = PEP^{-1}, where E = D^{1/2} =  \begin{bmatrix}5\sqrt2 & 0 \\ 0 & 10\sqrt2\end{bmatrix}.

    Having found R, you can then get U as U = AR^{-1}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: January 3rd 2012, 05:16 AM
  2. Crout algorithm to finding LU decomposition
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 20th 2010, 09:19 AM
  3. Jordan Decomposition to Schur Decomposition
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 30th 2009, 01:52 PM
  4. Finding polar form
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: November 23rd 2008, 07:26 PM
  5. Finding polar equations
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 3rd 2008, 06:58 PM

Search Tags


/mathhelpforum @mathhelpforum