Ex. 14.19

Ex. 14.19

If \(\bA\) is \(p\times p\) orthogonal, show that the first term in (14.92) on page 568

\[\begin{equation} \sum_{j=1}^p\sum_{i=1}^N\log(\phi(a_j^Tx_i)),\non \end{equation}\]

with \(a_j\) the \(j\)th column of \(\bA\), does not depend on \(\bA\).

Soln. 14.19

Since \(\bb{A}\) is orthogonal, we have

\[\begin{eqnarray} \sum_{j=1}^p\sum_{i=1}^N\log(\phi(a_j^Tx_i)) &=& \sum_{i=1}^N\sum_{j=1}^p\log(\phi(a_j^Tx_i))\non\\ &=&\sum_{i=1}^N(2\pi)^{-p/2}e^{-x_i^T\bb{A}\bb{A}^Tx_2/2}\non\\ &=&(2\pi)^{-p/2}\sum_{i=1}^Ne^{-x_i^Tx_2/2}\non \end{eqnarray}\]

does not depend on \(\bb{A}\).