The Missing Singular Value

A dataset has $n = 10$ samples and $4$ features. The columns have already been mean-centred, giving the $10 \times 4$ matrix $X$ , and the sample covariance matrix is

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

The per-feature sample variances — the diagonal of $S$ — are on record:

Feature	Sample variance
$f_1$	$5.0$
$f_2$	$4.0$
$f_3$	$3.0$
$f_4$	$2.0$

A separate job ran the SVD of $X$ and printed the singular values in decreasing order, $\sigma_1 > \sigma_2 > \sigma_3 > \sigma_4$ . The log file was truncated and only three survived:

Singular value	Logged
$\sigma_1$	$9$
$\sigma_2$	(lost)
$\sigma_3$	$4$
$\sigma_4$	$2$

PCA is then run on $S$ .

What percentage of the total variance is explained by the first two principal components?

Round your answer to 2 decimal places.

The Missing Singular Value

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