A two-variable analysis
For
S=\begin{pmatrix}3&1\\1&3\end{pmatrix},
find the principal component directions, the variances of the two components, and the proportion of variance explained by the first.
Worked answer (click to show)
The symmetric covariance treats the two coordinates equally, which suggests the sum and difference directions. The eigenvalue calculation confirms which direction has greater variance.
The characteristic equation is (3-\lambda)^2-1=0, so \lambda_1=4 and \lambda_2=2. Solving (S-4I)\mathbf{a}=\mathbf{0} gives a_1=a_2, so
\mathbf{a}_1=(1,1)^\top/\sqrt{2},
and the orthogonal direction is \mathbf{a}_2=(1,-1)^\top/\sqrt{2}. The component variances are 4 and 2, and the first explains
4/(4+2)=2/3\approx 66.7\%
of the total variance.