Least Squares derivation - vector commutative

Question

$ \frac{d}{dw} = \frac{1}{N} \sum_{n=1}^N (y_n - x_n^Tw)x_n + 2\lambda w $

$ = \frac{1}{N} \sum_{n=1}^N y_nx_n - x_n^Twx_n + 2 \lambda w$

can i shift w anyhow I like ? eg

$?= \frac{1}{N} \sum_{n=1}^N y_nx_n - x_nx_n^Tw + 2 \lambda w $

Is this correct ? Why yes or why no ?

Answer 1

If $x_n^Tw$ is a scalar, then we have

$$(x_n^Tw)x_n = x_n(x_n^Tw)$$

Additional remark:

• $\frac{d}{dw}$ is not relevant.

• try to use paranthesis reasonably generously.

$$\sum_{n=1}^N (y_n - x_n^Tw)x_n = \sum_{n=1}^N (y_nx_n - (x_n^Tw)x_n)$$