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Trace of Matrix Product as Scalar Product

  1. Aug 2, 2013 #1
    1. The problem statement, all variables and given/known data

    Let V be the real vector space of all real symmetric n × n matrices and define the scalar product of two matrices A, B by (Tr (A) denotes the trace of A)

    Show that this indeed fulfils the requirements on a scalar product.

    tracescalarproduct1.png

    2. Relevant equations

    Conditions for a scalar product:

    tracescalarproduct2.png

    3. The attempt at a solution

    I'm not sure how to show the last part. Which can be summarized as:

    <A|B> = 0 if ATA = I and BTB = I

    The first 3 parts of my attempt are shown below:

    tracescalarproduct3.png
     
  2. jcsd
  3. Aug 2, 2013 #2

    micromass

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    For (2), you say ##\mathrm{Tr}(AB)^T##. I don't really know what you mean with this. What is the transpose of a number? And why should

    [tex]\mathrm{Tr}(AB)^T = \mathrm{Tr}(B^T A^T)[/tex]

    For (3), you should still show that the inner product is ##\geq 0## and that it is ##=0## iff ##A=0##.
     
  4. Aug 3, 2013 #3
    For rule 1, I think the OP means [itex]Tr ((AB)^T)[/itex], not [itex](Tr(AB))^T[/itex]. The 1st line of the proof is unnecessary. The 2nd line looks good to me. However, you are using the fact that [itex]Tr(A) = Tr(A^T)[/itex]. This is easy to prove and you should add that proof in. For rule 2, you still need to prove that <A|A> = 0 implies A = 0. You will need to use the fact that the underlying space only includes symmetric matrices.
     
    Last edited: Aug 3, 2013
  5. Aug 5, 2013 #4
    Yeah [itex]Tr(A) = Tr(A^T)[/itex] because for any [itex]A_{ij}[/itex] component where i=j, switching their positions don't change anything.

    I'm more concerned about the point number 4. Which can be summarized as:

    <A|B> = 0 if ATA = I and BTB = I
     
  6. Aug 5, 2013 #5

    D H

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    Point number 4 says nothing of the kind. Consider A=B=I. Obviously ATA = I, as does BTB. Yet <A,B> is not zero.

    Instead think of point #4 as being a definition of what it means for two quantities to be deemed as being "orthogonal" to one another.
     
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