In short, I’m trying to go from the first line to the second line in this equation:
I feel foolish but on the second line my factor of $x_t$ is coming out transposed. Can someone please illuminate the steps so I can see where I went wrong? This is self study.
Since both ${\bf x}_t$ and ${\bf w}_t$ are $d$-dimensional column (?) vectors, the product ${\bf w}_t^T {\bf x}_{t}$ is a scalar, so it's equal to its own transpose: $${\bf w}_t^T {\bf x}_{t}=({\bf w}_t^T {\bf x}_{t})^T={\bf x}_t^T{\bf w}_t. $$ Therefore the last term on the first line of (9.9) can be written (ignoring the $\alpha$): $$-({\bf w}_t^T {\bf x}_{t}){\bf x}_t=-{\bf x}_t({\bf w}_t^T {\bf x}_{t})^T=-{\bf x}_t{\bf x}_t^T{\bf w}_t. $$ A similar calculation should work for the second to last term.