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Write trace of AB* as summation

  1. Mar 20, 2014 #1
    i'm kinda confused regarding summation so i'm hoping someone can help me figure this out and explain to me why it is the way it is

    trace(AB*) = ? in summation form

    * = adjoint = conjugate and transpose = transpose and conjugate

    assume both matrices are square mx of same size n x n

    trace = sum of diagonal entries

    i'm got this after brute force

    (summation of this entire thing) a_ij x conjugate of (b_ij)

    i, j runs from 1 to n


    but somehow i'm thinking it should be

    (summation of this entire thing) a_ii x conjugate of (b_ii)

    i runs from 1 to n
     
  2. jcsd
  3. Mar 21, 2014 #2

    Erland

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    A diagonal element aii of a matrix product AB depends on all elements aik of row i in A and all elements bki of column i in B, not just on the diagonal elements of A and B. Therefore, the trace of AB must be the sum of aikbki over all i and k, not just the sum of aiibii.
     
  4. Mar 21, 2014 #3

    Fredrik

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    It sounds like what you're confused about isn't summation, but rather the definition of matrix multiplication. I'll quote myself:

    The problem is quite easy if you just use the definitions. The other definitions you need to use are ##\operatorname{Tr X}=\sum_{i}X_{ii}## and ##(X^*)_{ij}=(X_{ji})^*##, where the first * denotes the adjoint operation and the second one denotes complex conjugation. But you don't seem to be confused about those.
     
    Last edited: Mar 21, 2014
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