= Solution
The calculation is <principal component analysis on a correlation matrix>, rather than on an unscaled <covariance matrix>. For each of the eleven rating variables, subtract its column mean and divide by its column <standard deviation>. If $Z$ is the resulting $43\times11$ <matrix>, use the divisor convention of the fitted object to write
$$
R=\frac1{43}Z^TZ,
\qquad R a_k=\lambda_k a_k,
\qquad a_k^Ta_\ell=\delta_{k\ell},
\qquad\lambda_1\ge\cdots\ge\lambda_{11}\ge0.
$$
Here the scaling <standard deviations> use divisor $43$, consistently with this convention. Using sample <standard deviations> with divisor $42$ gives the same <correlation matrix> and directions, with the corresponding score scale. Every diagonal entry of $R$ is one, so $\sum_k\lambda_k=\operatorname{tr}R=11$.
A <principal component> is the <linear combination> of the standardized ratings along $a_k$. Its <principal component score> for judge $i$ is $t_{ik}=z_i^Ta_k$, and its <variance> is $\lambda_k$. The displayed component <standard deviations> are $\sqrt{\lambda_k}$; the <explained variance of a principal component> is $\lambda_k/11$. In particular,
$$
\lambda_1\simeq(3.1833029)^2\simeq10.1334,\qquad
\lambda_2\simeq(0.65163561)^2\simeq0.424629.
$$
The first component explains about $92.122\%$ of the standardized variation, and the first two jointly explain \b[$95.982\%$]. Components three, four and five add about $2.321\%$, $0.830\%$ and $0.339\%$. The cumulative line adds these proportions successively; it is not the proportion explained by that component alone. Retaining two components discards about $4.018\%$ of the total standardized squared variation.
The displayed <principal component loadings> are the unit <eigenvector> coefficients $a_{jk}$. The first direction has positive, broadly similar weights for all ratings. It measures an overall favorable-rating tendency: a judge high on most ratings has a high first score. The second direction compares integrity and demeanour, with large positive coefficients, against such attributes as decision promptness, case-flow management and physical ability, with negative coefficients. Thus among similarly rated judges, a positive second score indicates relatively stronger integrity/demeanour compared with those operational ratings. This is a description of a data direction, not a causal interpretation. Multiplying a whole <eigenvector> and its scores by minus one reflects the display but changes nothing substantive.
These <principal component loadings> are not themselves variable-component correlations. With standardized variables,
$$
\operatorname{Cov}(Z_j,a_k^TZ)=\lambda_k a_{jk},
\qquad\operatorname{Corr}(Z_j,a_k^TZ)=\sqrt{\lambda_k}\,a_{jk}.
$$
For example, the integrity correlation with component one is approximately $3.1833\times0.289\simeq0.920$. Blank loadings in the printed second column indicate suppressed small entries, not exact zeros. The complete unrounded loading vectors in the fitted object must be used to calculate scores.
A two-dimensional representation plots the 43 pairs
$$
\boxed{(t_{i1},t_{i2})=(z_i^Ta_1,z_i^Ta_2),\qquad i=1,\ldots,43,}
$$
with each point labeled by its judge. In the fitted software object these are the first two columns of the score <matrix>. Nearby points have similar projections of their standardized ratings. The rank-two reconstruction $\widehat z_i=t_{i1}a_1+t_{i2}a_2$ is the <orthogonal projection> onto the leading principal plane and minimizes the total squared reconstruction error over all two-dimensional linear subspaces. To see this, for any orthonormal pair of directions the captured <variance> is the sum of its two Rayleigh quotients, at most $\lambda_1+\lambda_2$ by diagonalizing $R$; the remaining <variance> is the total minus that captured sum. Distances in the plot omit the smaller components, so a close projected pair need not be equally close in every original rating. The actual 43 coordinates cannot be recovered from the summary alone; the supplied data <matrix> or fitted score <matrix> is needed.
The printed descriptive list omits INTG, although the loading table includes it. INTG denotes integrity and supplies the eleventh variable. The standard data set also contains a contacts variable; the stated eleven-rating analysis uses the ratings, not contacts.
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