Results 1 to 5 of 5

Math Help - Matrix of a bilinear form

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    62

    Matrix of a bilinear form

    Hi,

    I've some troubles finding a matrix of a bilinear form:

    Given n >= 1, n in IN and beta : M_{n}(\mathbf{R}) X M_{n}(\mathbf{R}) ----> IR the application defined by beta( A,B)=Tr({t}^A,B) \forall A,B \in M_{n}(\mathbf{R}
    (IN= positive integers, IR= real numbers).

    Find the matrix of beta in the standar basis of M_{n}(\mathbf{R} .

    So I'm stuck here because I don't see what kind of n by n matrix would give me

    A*(matrix)*B=\beta (A,B).



    I am really sorry the Latex compiler doesn't seem to work .
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,150
    Thanks
    591
    have you tried computing β on basis elements of Mn(R), considered as elements of R^(n^2)? for example, in M2(R), we can identify

    [0 0]
    [1 0] with e3 in R^4.

    you should get an n^2 x n^2 matrix, since dim(Mn(R)) = n^2.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    44
    Denote B = { E_{ ij } } the standard basis in M_n ( IR ) . Prove that

    beta ( E_{ ij} , E_{ ij} ) = 1

    beta ( E_{ ij} , E_{ kh} ) = 0 if ( i , j ) =/= ( k , h )

    As a consequence, the matrix of beta in the standard basis of M_n ( IR ) is I_{ n^2} .
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,150
    Thanks
    591
    the seaside opening of the river kronecker rises again
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    44
    Quote Originally Posted by Deveno View Post
    the seaside opening of the river kronecker rises again

    Hilbert would be sad.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. OLS in matrix form
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: August 31st 2010, 07:50 AM
  2. matrix canonical form
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 29th 2009, 09:37 AM
  3. matrix row echelon form
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 17th 2008, 04:23 PM
  4. Row-echelon form of a matrix
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 26th 2008, 09:51 AM
  5. bilinear form
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: August 24th 2007, 06:14 AM

Search Tags


/mathhelpforum @mathhelpforum