Homework for Periodic Systems

  1. (AntiHermitian)

    Show that the eigenvalues of anti-Hermitian matrices are pure imaginary.

  2. (TwoMatrices)

    Consider the following two matrices:

    \begin{align} S &= \left[ \begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ \end{array} \right] \\ A &= \left[ \begin{array}{rrrrr} 2 & i & 0 & 0 &-i \\ -i & 2 & i & 0 & 0 \\ 0 &-i & 2 & i & 0 \\ 0 & 0 &-i & 2 & i \\ i & 0 & 0 &-i & 2 \\ \end{array} \right] \end{align}

    1. Explain the effect of matrix $S$ on an arbitrary column vector $v$. \emph{It may help to invent a vector $v$ and multiply it by $S$.}

    2. Show that these two matrices commute.

    3. Find the eigenstates and eigenvalues of $S$.

    4. Demonstrate that the eigenstates of $S$ are also eigenstates of $A$, and find the corresponding eigenvalues.


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