Homework for Commuting Matrices

  1. (Pauli) This short exercise introduces the terms “commutator” and “anticommutator”. The context is Pauli spin matrices. It can be used in conjunction with the homework problem (PauliPractice).

    The Pauli spin matrices $\sigma_x$, $\sigma_y$, and $\sigma_z$ are defined by:

    $$\sigma_x=\pmatrix{0&1\cr 1&0\cr}\qquad\qquad \sigma_y=\pmatrix{0&-i\cr i&0\cr}\qquad\qquad \sigma_z=\pmatrix{1&0\cr 0&-1\cr}$$

    These matrices are related to angular momentum in quantum mechanics. Prove, and become familiar with, the identities listed below.

    1. Show that each of the Pauli matrices is hermitian. (A matrix is hermitian if it is equal to its hermitian adjoint.

    2. Show that the determinant of each of the Pauli matrices is $-1$.

    3. Show that $\sigma_i^2={\cal I}$ for each of the Pauli matrices, i.e.\ for $i\in\left\{x,y,z\right\}$.

  2. (Commutator) This was intended to be a short exercise to encourage students to look at the geometric meaning of matrices that commute. We have only used it once and the (very sophisticated) TA spent forever on it, trying to explain what the matrices do. Use it at your own risk.

    Consider the following matrices:

    $$A=\pmatrix{0&1\cr 1&0\cr}\qquad\qquad B=\pmatrix{3&1\cr 1&3\cr}\qquad\qquad C=\pmatrix{1&0\cr 0&-1\cr}$$

    1. Explain what each of the matrices “does” geometrically when thought of as a linear transformation acting on a vector.

    2. The commutator of two matrices $A$ and $B$ is defined by $\left[A, B\right]\buildrel \rm def \over = AB-BA$. Find the following commutators: $\left[A,B\right]$, $\left[A,C\right]$, $\left[B,C\right]$.

    3. Two matrices are said to “commute,” if their commutator is zero. Thought of as linear transformations, two matrices commute if it doesn't matter in which order the transformations act. For all pairs of the matrices $A$, $B$, and $C$, show geometrically that the order of the transformations doesn't matter when the matrices commute and does matter when they don't commute.

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