## Section6.8Matrix Examples

Any $2\times2$ complex matrix, $M\text{,}$ can be written in the form

$$M = \begin{pmatrix} t+z\amp x-iy\\ x+iy\amp t-z \end{pmatrix}\tag{6.8.1}$$

with $x,y,z,t\in\RR\text{.}$ Looking at the matrix coefficients of these variables, we can write

$$M = t \,I + x \,\sigma_x + y \,\sigma_y + z \,\sigma_z\tag{6.8.2}$$

thus defining the three matrices

\begin{align} \sigma_x \amp = \begin{pmatrix} 0\amp 1\\ 1\amp 0 \end{pmatrix} ,\tag{6.8.3}\\ \sigma_y \amp = \begin{pmatrix} 0\amp -i\\ i\amp 0 \end{pmatrix} ,\tag{6.8.4}\\ \sigma_z \amp = \begin{pmatrix} 1\amp 0\\ 0\amp -1 \end{pmatrix}\text{,}\tag{6.8.5} \end{align}

which are known as the Pauli matrices. The Pauli matrices $\{\sigma_m\}$ have several interesting properties. First of all, each Pauli matrix squares to the identity matrix, is tracefree, and of course is Hermitian:

\begin{align} \sigma_m^2 \amp = I ,\tag{6.8.6}\\ \tr(\sigma_m) \amp = 0 ,\tag{6.8.7}\\ \sigma_m^\dagger \amp = \sigma_m\text{.}\tag{6.8.8} \end{align}

Also of interest is that these matrices anticommute, and that the product of any two is the third, for instance:

$$\sigma_x \sigma_y = i\,\sigma_z = -\sigma_y \sigma_x\text{.}\tag{6.8.9}$$

Thus, any $2\times2$ Hermitian matrix is a linear combination of the Pauli matrices and the identity matrix. What about $2\times2$ unitary matrices?

Now we must have $MM^\dagger=I\text{,}$ that is,

$$\begin{pmatrix} a\amp b\\ c\amp d \end{pmatrix} \begin{pmatrix} a^*\amp c^*\\ b^*\amp d^* \end{pmatrix} = \begin{pmatrix} |a|^2+|b|^2\amp ac^*+bd^*\\ ca^*+db^*\amp |c|^2+|d|^2 \end{pmatrix} = I\text{.}\tag{6.8.10}$$

The normalization conditions

$$|a|^2+|b|^2 = 1 = |c|^2+|d|^2\tag{6.8.11}$$

allow us to write

\begin{align} a \amp = e^{i\alpha}\cos\theta ,\tag{6.8.12}\\ b \amp = e^{i\beta}\sin\theta ,\tag{6.8.13}\\ c \amp = e^{i\gamma}\sin\phi ,\tag{6.8.14}\\ d \amp = e^{i\delta}\cos\phi\text{,}\tag{6.8.15} \end{align}

where we can assume that $\theta,\phi\in[0,\frac{\pi}{2}]\text{.}$ The remaining condition becomes

$$0 = ac^*+bd^* = e^{i(\alpha-\gamma)}\cos\theta\sin\phi + e^{i(\beta-\delta)}\sin\theta\cos\phi\tag{6.8.16}$$

which forces in general

\begin{align} \tan\theta \amp = \tan\phi ,\tag{6.8.17}\\ e^{i(\alpha-\gamma)} \amp = -e^{i(\beta-\delta)}\tag{6.8.18} \end{align}

(with the second condition unnecessary if $\theta=0$ or $\theta=\frac{\pi}{2}$), so that

\begin{align} \theta \amp = \phi ,\tag{6.8.19}\\ \alpha + \delta \amp = \beta + \gamma - \pi\text{.}\tag{6.8.20} \end{align}

Some special cases are $\alpha=0=\delta\text{,}$ $\beta=\frac{\pi}{2}=\gamma\text{,}$ corresponding to

$$U_x = \begin{pmatrix} \cos\theta\amp i\sin\theta\\ i\sin\theta\amp \cos\theta \end{pmatrix}\text{,}\tag{6.8.21}$$

$\alpha=0=\delta\text{,}$ $\beta=0\text{,}$ $\gamma=\pi\text{,}$ corresponding to

$$U_y = \begin{pmatrix} \cos\theta\amp \sin\theta\\ -\sin\theta\amp \cos\theta \end{pmatrix}\text{,}\tag{6.8.22}$$

and $\theta=0\text{,}$ $\delta=\alpha\text{,}$ corresponding to

$$U_z = \begin{pmatrix} e^{i\theta}\amp 0\\ 0\amp e^{-i\theta} \end{pmatrix}\text{,}\tag{6.8.23}$$

where we have replaced $\alpha$ by $\theta$ for consistency.