Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 35 / 1 / a / Solution

Write $n=bk$, let $I_n$ be the identity matrix, let $J_m$ be the $m$-by-$m$ all-ones matrix, and put $B=\mathrm{diag}(J_k,\dots ,J_k)$, with one diagonal block per experimental block. The variance-covariance matrix isIts orthogonal decomposition is obtained by first subtracting each block's sample mean, then subtracting the grand sample mean from the block means. More explicitly, writing $x_{jt}$ for coordinate $t$ in block $j$, the three invariant subspaces areThey are pairwise orthogonal, with dimensions $b(k-1)$, $b-1$ and $1$, and their direct sum is ${\mathbb{R}}^n$. On $W$, both $B$ and $J_n$ vanish. On $U$, $B=kI$ and $J_n=0$. On $G$, $B=kI$ and $J_n=nI$. Thus the eigenvalues and corresponding invariant subspaces areIf some eigenvalues coincide, their eigenspace is the direct sum of the listed subspaces with that value. Zero-dimensional rows are omitted when $b=1$ or $k=1$. For an admissible covariance matrix the eigenvalues on nonzero subspaces must be nonnegative; these conditions are also sufficient for positive semidefiniteness. The expectation parameters do not affect this calculation.