1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sum of Unitary Matrices Question

  1. Sep 20, 2015 #1

    RJLiberator

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Find an example of two unitary matrices that when summed together are not unitary.

    2. Relevant equations


    3. The attempt at a solution

    A = \begin{pmatrix}
    0 & -i\\
    i & 0\\
    \end{pmatrix}

    B = \begin{pmatrix}
    0 & 1\\
    1 & 0\\
    \end{pmatrix}

    A+B =
    A = \begin{pmatrix}
    0 & 1-i\\
    1+i & 0\\
    \end{pmatrix}

    So we see that the hermitian conjugate of (A+B) is identical to A+B.

    So (A+B)(A+B) =
    A = \begin{pmatrix}
    2 & 0\\
    0 & 2\\
    \end{pmatrix}

    So since it is a diagonal matrix of 2, this is not the identity matrix. We can safely conclude that while A is unitary, B is unitary, (A+B) is NOT unitary.

    Safe understanding?

    Thanks
     
  2. jcsd
  3. Sep 21, 2015 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    That's fine. Even simpler, if ##I## is the identity matrix, then ##I## is unitary, so is ##-I##. ##I+(-I)=0##. ##0## is not unitary.
     
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: Sum of Unitary Matrices Question
  1. Unitary matrices! ! (Replies: 10)

  2. Unitary Matrices Problem (Replies: 20)

  3. Unitary Matrices (Replies: 20)

Loading...