Problem 1.1. Let \(A\) be partitioned as \(A = \begin {pmatrix} A_{11} & A_{12}\\ A_{21} & A_{22}\end {pmatrix}\) with \(A_{11}\) non-singular. Show that \[\left |A\right | = \left |A_{11}\right |\, \left |A_{22}-A_{21}A_{11}^{-1}A_{12}\right |.\] Where to start: write \(A = BD\) with \(B\) block lower triangular and \(D\) block upper triangular, and use \(|BD| = |B||D|\). The second factor is the Schur complement, which reappears as a conditional covariance in Section 5.1.

Problem 1.2. Prove that \(\operatorname {tr}(AB) = \operatorname {tr}(BA)\) whenever both products are defined, and deduce that \(\operatorname {tr}(P\Lambda P') = \operatorname {tr}(\Lambda )\) for orthogonal \(P\). This is the identity behind Theorem 4.6.

Problem 1.3. The centring matrix is \(H = I - \tfrac {1}{n}\underline {1}\,\underline {1}'\). Show that \(H\) is symmetric and idempotent, that \(H\underline {1}=\underline {0}\), and that \(\operatorname {tr}(H) = n-1\). Explain what each of these three facts says about what \(H\) does to a data matrix.

Problem 1.4. Show that a symmetric matrix \(A\) is positive definite if and only if every eigenvalue is strictly positive, and that \(\left |A\right | = \prod \lambda _i\).

Problem 1.5. Obtain the spectral decomposition of \(S = \begin {pmatrix} 4 & 1\\ 1 & 4\end {pmatrix}\), and verify both \(\left |S\right | = \lambda _1\lambda _2\) and \(\operatorname {tr}(S) = \lambda _1+\lambda _2\).

Problem 1.6. For \(\Sigma \) positive definite, define the symmetric square root \(\Sigma ^{1/2} = P\Lambda ^{1/2}P'\) and verify that \(\Sigma ^{1/2}\Sigma ^{1/2}=\Sigma \) and that \(\Sigma ^{1/2}\) is symmetric. Where is this matrix used later in the course?

Problem 1.7. Show that if \(A\) is \(n\times p\) with \(n<p\) then \(A'A\) is singular. Relate this to the statement in Section 5.3 that a multivariate analysis requires more observations than variables.