Bayesian regression kernel

Question

I am reading Bishops book "pattern recognition and machine learning". In chapter 3 "linear models for regression" section 3.3.3 "equivalent kernel" equation 3.63 on page 160 is given as follows:

$$cov[y(x),y(x')]=cov[\phi(x)^Tw,w^T\phi(x')]=\phi(x)^TS_N\phi(x')=\beta^{-1}k(x,x')$$

where $x$ is a data vector, $\phi$ is a vector of basis functions, $t$ the target values, $w$ the regression weights, $y(x)=\phi(x)^Tw$, and the posterior of $w$ is given by $p(w|t)\sim N(w;m_N,S_N)$

I do not understand the middle part of the equation given. How is this step below derived?

$$cov[\phi(x)^Tw,w^T\phi(x')]=\phi(x)^TS_N\phi(x')$$

Answer 1

Figured it out. Didn't see the obvious before that $\phi(x)$ is constant:

$$cov[\phi(x)^Tw,w^T\phi(x')]=\mathbb{E}[(\phi(x)^Tw-\mathbb{E}[\phi(x)^Tw])(w^T\phi(x')-\mathbb{E}[w^T\phi(x')])]=\mathbb{E}[\phi(x)^T(w-\mathbb{E}[w])(w^T-\mathbb{E}[w^T])\phi(x')]=\phi(x)^T \mathbb{E}[(w-\mathbb{E}[w])(w^T-\mathbb{E}[w^T])] \phi(x')= \phi(x)^T S_N \phi(x')$$