Let
\[|\alpha\rangle \doteq \frac{1}{\sqrt{2}}
\begin{pmatrix}
1\\ 1
\end{pmatrix}
\qquad \rm{and} \qquad
|\beta\rangle \doteq \frac{1}{\sqrt{2}}
\begin{pmatrix}
1\\ -1
\end{pmatrix}\]
Show that \(\left|{\alpha}\right\rangle \) and \(\left|{\beta}\right\rangle \) are orthonormal.
(If a pair of vectors is orthonormal, that suggests that
they might make a good basis.)

Consider the matrix
\[C\doteq
\begin{pmatrix}
3 & 1 \\ 1 & 3
\end{pmatrix}
\]
Show that the vectors
\(|\alpha\rangle\) and
\(|\beta\rangle\) are
eigenvectors of C and find the eigenvalues.
(Note that showing something is an eigenvector of an operator is far easier than finding the eigenvectors if you don't know them!)

A operator is always represented by a diagonal matrix if it is written in terms of
the basis of its own eigenvectors. What does this mean? Find the matrix elements for a
new matrix \(E\) that
corresponds to \(C\) expanded in the basis of its eigenvectors, i.e. calculate \(\langle\alpha|C|\alpha\rangle\),
\(\langle\alpha|C|\beta\rangle\), \(\langle\beta|C|\alpha\rangle\) and
\(\langle\beta|C|\beta\rangle\)
and arrange them into a sensible matrix \(E\). Explain why you arranged the matrix
elements in the order that you did.

Find the determinants of \(C\) and \(E\). How do these determinants compare to the eigenvalues of these matrices?