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Math Help - Matrix Exponential Determinant

  1. #1
    Pur
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    Matrix Exponential Determinant

    Prove: The determinant of the matrix exponential of A is e^{tr{A}}, i.e:
    \det{e^{A}}=e^{tr{A}}

    Thoughts: Look at the eigenvalues...
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  2. #2
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    Quote Originally Posted by Pur View Post
    Prove: The determinant of the matrix exponential of A is e^{tr{A}}, i.e:
    \det{e^{A}}=e^{tr{A}}

    Thoughts: Look at the eigenvalues...
    That's right! Look at the eigenvalues.

    If \lambda is an eigenvalue of A with eigenvector x, then A^nx = \lambda^n x for all n, and hence e^Ax = e^\lambda x. So the eigenvalues of e^A are the exponentials of the eigenvalues of A; and the determinant in each case is the product of the eigenvalues.

    Edit. Unfortunately, that answer was incomplete, because it does not deal with multiplicity. You need the matrices A and e^A not only to have the same eigenvalues, but to have them with the same multiplicity. One way to do this is to see that the Jordan normal form of e^A has the exponentials of the eigenvalues of A as its diagonal elements. In the answer I gave above, I was trying to avoid having to use the Jordan normal form, but maybe it's unavoidable.
    Last edited by Opalg; April 10th 2009 at 01:57 AM. Reason: Answer was incomplete
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