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Homework Help: Sum of Unitary Matrices Question

  1. Sep 20, 2015 #1


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    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\\

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

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

    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\\

    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?

  2. jcsd
  3. Sep 21, 2015 #2


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    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.
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