Results 1 to 9 of 9

Math Help - An identity of Jacobi

  1. #1
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1

    An identity of Jacobi

    Show that, if A is any complex-valued matrix,

    \mbox{det }e^A = e^{Tr(A)}

    where Tr is the trace.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    ynj
    ynj is offline
    Senior Member
    Joined
    Jul 2009
    Posts
    254
    Quote Originally Posted by Bruno J. View Post
    Show that, if A is any complex-valued matrix,

    \mbox{det }e^A = e^{Tr(A)}


    where Tr is the trace.

    where is the variable of A?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    A is a matrix with complex entries.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    ynj
    ynj is offline
    Senior Member
    Joined
    Jul 2009
    Posts
    254
    Quote Originally Posted by Bruno J. View Post
    A is a matrix with complex entries.
    I am confused..how can you calculate the exponent of a matrix....
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Follow Math Help Forum on Facebook and Google+

  6. #6
    ynj
    ynj is offline
    Senior Member
    Joined
    Jul 2009
    Posts
    254
    Let fbe a polynomial,if the eigenvalue of Ais \lambda_1,...,\lambda_n, then the eigenvalue of f(A)is f(\lambda_1),...,f(\lambda_n)
    we may look e^Aas a polynomial series of A(limit)
    note that: \det A=\lambda_1\lambda_2...\lambda_n,Tr(A)=\lambda_1+\  lambda_2...+\lambda_n
    e^{Tr(A)}=e^{\lambda_1}e^{\lambda_2}...e^{\lambda_  n}
    but \{e^{\lambda_i}\}is the eigenvalue of e^A(regard it as polynomial)
    so \det e^A=e^{\lambda_1}e^{\lambda_2}...e^{\lambda_n}=e^{  Tr(A)}
    Last edited by ynj; August 18th 2009 at 05:08 PM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Remember that not every matrix is diagonalizable!

    In the case where the matrix is diagonal then it is easy to show that the identity holds because if a is one of the diagonal entries, the corresponding entry of e^A will be e^a.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    ynj
    ynj is offline
    Senior Member
    Joined
    Jul 2009
    Posts
    254
    I have changed my proof
    every matrix have eigenvalue, and my proof is not based on diagonalibility
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Bruno J. View Post
    Show that, if A is any complex-valued matrix,

    \mbox{det }e^A = e^{Tr(A)}

    where Tr is the trace.
    See here (2/3rd of the way down)

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Jacobi Symbol
    Posted in the Number Theory Forum
    Replies: 12
    Last Post: March 8th 2010, 08:53 PM
  2. Jacobi identity
    Posted in the Math Challenge Problems Forum
    Replies: 11
    Last Post: October 8th 2009, 12:20 AM
  3. [SOLVED] Jacobi Symbol
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: December 13th 2008, 07:52 PM
  4. Jacobi Symbol
    Posted in the Number Theory Forum
    Replies: 6
    Last Post: May 1st 2008, 07:04 PM
  5. Jacobi identity
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: August 6th 2006, 08:08 PM

Search Tags


/mathhelpforum @mathhelpforum