Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A Question about properites of tensor product

  1. May 4, 2016 #1

    td21

    User Avatar
    Gold Member

    They are being 2 by 2 matrices and I being the identity. Physically they are Pauli matrices.

    1. Is $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes B$$

    = $$(A\otimes I\otimes I)\otimes B + (I\otimes A\otimes I)\otimes B + (I\otimes I\otimes A)\otimes B$$? I think so.

    2. Is $$(A\otimes I\otimes I)\otimes B + (I\otimes A\otimes I)\otimes B + (I\otimes I\otimes A)\otimes B$$=

    $$(A\otimes B)\otimes (I\otimes I)\otimes (I\otimes I) + (I\otimes I)\otimes (A\otimes B)\otimes (I\otimes I) + (I\otimes I)\otimes(I\otimes I)\otimes(A\otimes B)$$? I am not sure.

    Note that they are no scalar product here. I ask 2 because I stumbled upon the RHS of 2 and hope to know if it can be factored out as the LHS of 1. So basically I reverse the line of reason to ask these two questions. Looking at the LHS of 1., I also want to know:

    3. Is $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes B$$
    = $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes (B\otimes I\otimes I)$$
    = $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes (I\otimes B\otimes I)$$
    =$$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes (I\otimes I\otimes B)$$? I do not think so as I do not think I can enlarge the dimension of B here?
    This is not homework.
     
  2. jcsd
  3. May 5, 2016 #2

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    1 is correct.
    2 is not, because the terms on the LHS are order-4 tensors and those on the RHS are order-6 tensors. If the atomic elements all have the same order then each term has to have the same number of ##\otimes## symbols in it.
    3 is not correct, for the same reason.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Question about properites of tensor product
Loading...