Results 1 to 4 of 4

Math Help - Derivative Problem

  1. #1
    Junior Member
    Joined
    Nov 2009
    From
    Pocatello, ID
    Posts
    59

    Derivative Problem

    Let E be the set of all 2n x 2n matrices of the form C=\left( {\begin{array}{cc}<br />
 A & -B  \\<br />
 B & A  \\<br />
 \end{array} } \right), for A, B\in Mat(n\times n, \mathbb{R}), and let G=\{C\in E : CC^T=I\}. Further, let f(C)=CC^T. This maps E into a subspace W=\{P\in E : P^T = P\}.

    Compute the derivative (Df)_C and show it's surjective when C\in G.

    So I computed the derivative and got (Df)_C(U)=CU^T+UC^T. For C\in G, this becomes CU^T+UC^{-1}. So for P\in W, somehow I need to come up with a matrix U\in E such that (Df)_C(U)=P, but I'm not seeing it at all.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32
    Put U := \frac 12 PC for P\inW.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2009
    From
    Pocatello, ID
    Posts
    59
    Following up on this problem, it says what I've just proven and the implicit function theorem imply that G is a manifold, and to determine its dimension. How do you go about determining the dimension of a manifold?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32
    Did you find the dimension of W ?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. derivative problem
    Posted in the Calculus Forum
    Replies: 4
    Last Post: September 25th 2010, 08:33 PM
  2. derivative problem -
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 2nd 2008, 11:09 AM
  3. Another derivative problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 26th 2008, 04:29 AM
  4. another derivative problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 25th 2008, 11:29 PM
  5. Derivative Problem
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 17th 2007, 07:12 PM

Search Tags


/mathhelpforum @mathhelpforum