Consider the properties of a \(2 \times 2\) real symmetric GOE random matrix of the form

\[\boldsymbol{R}=\left[\begin{array}{ll}x & z \\z & y\end{array}\right]\]

where the symmetric off-diagonal element \(z\) is selected from a Gaussian distribution with zero mean and unit variance, and diagonal elements \(x\) and \(y\) are selected from Gaussian distribution with zero mean and variance \(\sigma^{2}=2\).

(a) Show that the average of the two eigenvalues is zero, consistent with equation (5.6.16a).

(b) Show that the average of the square of the two eigenvalues is 3 , consistent with equation (5.6.16b).

(c) Show that the joint eigenvalue distribution function is given by \[ho\left(\lambda_{1}, \lambda_{2}\right)=\frac{1}{8 \sqrt{2 \pi}}\left|\lambda_{1}-\lambda_{2}\right| \exp \left(-\frac{\lambda_{1}^{2}+\lambda_{2}^{2}}{4}\right)\]
consistent with equation (5.6.18).

(d) Show that the distribution of the eigenvalue spacing is given by Wigner's surmise in equation (5.6.19).

(e) Show that the probability distribution of the eigenvalues is given by \[P(\lambda)=\frac{1}{4 \sqrt{2 \pi}}\left(2 e^{-\lambda^{2} / 2}+\sqrt{\pi} e^{-\lambda^{2} / 4} \lambda \operatorname{erf}(\lambda / 2)\right)\]
which is symmetric about zero, and has variance \(\sigma^{2}=3\).