= Solution
The command centres the log measurements and performs <principal component analysis> of their <covariance matrix>. With `cor=F`, it does not first standardize the variables to variance one. For a unit <principal component loading> vector $q$, the score $q^T(z_i-\bar z)$ has <sample variance> $q^T\widehat\Sigma q$. By the <spectral theorem for real symmetric matrices>, expand $q$ in an <orthonormal eigenbasis>; that <variance> is a weighted average of the <eigenvalues>. Its maximum is the largest <eigenvalue>, reached at the corresponding <eigenvector>. Repeating subject to <orthogonality> to previous directions gives the other <principal components>.
The software uses the divisor-$31$ <sample covariance matrix> for its component standard deviations, whereas `var` used divisor $30$. Diagonalizing the displayed matrix gives approximately
$$
\lambda(\widehat\Sigma)=(0.3323932,\ 0.00552536,\ 0.000970632),
\qquad
\lambda(S_{31})=\frac{30}{31}\lambda(\widehat\Sigma).
$$
Taking square roots of the latter gives $(0.5671603,0.07312405,0.03064834)$, agreeing with the output to the rounding of the displayed <sample covariance matrix>. The normalization changes these numerical standard deviations but leaves the <eigenvectors> and proportions of <explained variance of a principal component> unchanged.
The first <principal component> explains \b[about $98.08\%$ of the total log-variable variance], the second another $1.63\%$, and the last about $0.286\%$. Keeping two preserves about $99.714\%$. An original <scree plot> displays the sharp drop after the first component:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-42-pca-scree.png]
{title=Principal-component variances and cumulative explained variance for the log-tree measurements}
Consequently the centered cloud is very elongated along one direction: a one-dimensional approximation represents almost all of its squared variation. From the displayed <covariance matrix>, a unit first <eigenvector>, with its arbitrary sign chosen positive, is approximately
$$
q_1=(0.3980,\ 0.09514,\ 0.9124)^T.
$$
Its <principal component score> is therefore $0.3980(\log G_i-\overline{\log G})+0.09514(\log H_i-\overline{\log H})+0.9124(\log V_i-\overline{\log V})$. All coefficients have the same sign, so this direction describes overall tree size, with log volume dominating the unstandardized variation. The summary alone supplies proportions, not the loading directions; the displayed <sample covariance matrix> supplies the latter. This is a geometric approximation, not evidence of an exact deterministic relation or independent original variables.
One further use is <principal component analysis on a correlation matrix>, obtained with `cor=T`, which equalizes marginal variances and answers a different scaling-sensitive question. Another is <principal component regression>: fit a response to selected <principal component scores> rather than highly correlated original predictors, choosing the number using prediction assessment. The fitted transformation also projects new measurements into the same low-dimensional coordinates, provided the training centering and scaling are retained.
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