The matrix $S$ is the <sample covariance matrix>
$$
S=\frac1n\sum_{r=1}^n(x_r-\bar x)(x_r-\bar x)^T,
\qquad \bar x=\frac1n\sum_{r=1}^nx_r.
$$
Changing $n$ to $n-1$ is another convention, but the objective must use the same convention throughout.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 205 / 2 / a / Solution

{title2=$S$}

For observations $x_1,\ldots,x_n$, the sample covariance matrix is $n^{-1}\sum_i(x_i-\bar x)(x_i-\bar x)^T$, or the same sum divided by $n-1$ under the unbiased convention.


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