Homework for Periodic Systems

  1. (AntiHermitian)

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

  2. (BasisTrans)

    Consider a (generic) $3\times 3$ Hermitian matrix $M$ with three orthonormal eigenvectors $\vert u\rangle$, $\vert v\rangle$, $\vert w\rangle$ with eigenvalues $\lambda_u$, $\lambda_v$,and $\lambda_w$, respectively. Suppose that you know $M$ and its basis vectors in a particular representation, so that $M$ is a particular matrix and the basis vectors are (known) columns. Build the matrix $U$ out of the columns that represent the basis vectors: \begin{equation} U=\big(\vert u\rangle\quad \vert v\rangle\quad \vert w\rangle\big) \end{equation}

    1. Show that $U$ is unitary, i.e.\ show that $U^{\dagger}=U^{-1}$.

    2. Show that $U^{\dagger} M U$ is a diagonal matrix and that the diagonal entries are the eigenvalues of $M$. (You will show that any matrix is diagonal in its own eigenbasis.)


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