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Trace and endomorphism

  1. Oct 9, 2013 #1
    Consider the 4x4 matrices
    A =
    (1 2 3 4)
    (5 6 7 8)
    (9 10 11 12)
    (13 14 15 16)


    B=
    (1 2 3 4)
    (8 5 6 7)
    (11 12 9 10)
    (14 15 16 13)

    The question I was asked was the following: Show that there does not exist an endomorphism f: ℝ4 -> ℝ4 and basis 'a' and 'b' of R^4, such that A = a[f]a and B=b[f]b.

    I have read in my notes and found that if the traces of the two matrices are not the same then they cannot represent the same endomorphism.

    I am struggling to see the intuition behind this though.

    Can anyone shed some light?

    Many thanks
     
  2. jcsd
  3. Oct 9, 2013 #2

    Office_Shredder

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    Staff Emeritus
    Science Advisor
    Gold Member

    The trace of a linear transformation is the sum of the eigenvalues of the matrix, and so is independent of the choice of basis.

    Alternatively, you can use the fact that Tr(AB) = Tr(BA) to show that if you conjugate a matrix by another matrix the trace is unchanged.
     
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