From Singular Values to Explained Variance

A dataset has $n = 26$ samples and $d = 4$ features. The data matrix $X$ is $26 \times 4$ and has already been mean-centred (every column sums to zero).

Its singular value decomposition $X = U\Sigma V^\top$ gives singular values

$\sigma_1 = 12, \qquad \sigma_2 = 6, \qquad \sigma_3 = 4, \qquad \sigma_4 = 2$

You run PCA using the sample covariance matrix

$S = \frac{1}{n-1}X^\top X$

What is the largest eigenvalue $\lambda_1$ of $S$, and what is the smallest number of principal components $k$ you must keep to retain at least $85\%$ of the total variance?