Problem 3.1. For a data matrix \(X\) of \(n\) observations on \(p\) variables, show that \[\overline {\underline {x}} = \tfrac {1}{n}X'\underline {1}, \qquad S = \tfrac {1}{n-1}X'HX,\] where \(H\) is the centring matrix. Where to start: \(HX\) has the column means subtracted from every entry.

Problem 3.2. The generalised variance is \(\left |S\right |\) and the total variance is \(\operatorname {tr}(S)\). Compute both for \(S = \begin {pmatrix} 4 & 3\\ 3 & 4\end {pmatrix}\) and for \(S = \begin {pmatrix} 4 & 0\\ 0 & 4\end {pmatrix}\). The total variance is the same in each; explain what the generalised variance detects that it does not.

Problem 3.3. Show that \(\left |S\right | = \prod ^{p}_{i=1}\lambda _i\) where the \(\lambda _i\) are the eigenvalues of \(S\), and hence that a zero generalised variance means at least one exact linear relation among the variables.

Problem 3.4. Data are recorded on height in centimetres and mass in kilogrammes. State what happens to \(\overline {\underline {x}}\), \(S\), \(R\), \(\operatorname {tr}(S)\) and \(\left |S\right |\) if height is re-expressed in metres. Which of these are invariant, and why does the answer matter for Section 4.3?

Problem 3.5. Show that the sample correlation between two variables is unchanged by any linear rescaling \(x\mapsto ax+b\) with \(a>0\), and reverses sign if \(a<0\).