I want to understand the answer to the following matrix equation but lack some knowledge of the linear algebra:
$(\beta-b)'B^{-1}(\beta-b)=(B^{\frac{-1}{2}}b-B^{\frac{-1}{2}}\beta)'(B^{\frac{-1}{2}}b-B^{\frac{-1}{2}}\beta)$
I am supposed to arrive with the result on RHS using Choleski decomposition which yields $B^{-1}=B^{\frac{-1}{2}'}B^{\frac{-1}{2}}$. The thing that really bugs me is the reversed order of $b$ and $\beta$ .
The above problem arises when I want to derive the marginals for the following posterior density: \begin{equation} p(\beta,\sigma^2|y) \propto \sigma^{-k} \exp\Bigg(\frac{-(\beta-b)'B^{-1}(\beta-b)}{2\sigma^2}\Bigg) \Bigg(\frac{1}{\sigma}\Bigg)^{N+2} \exp \Bigg(\frac{-(y-X\beta)'(y-X\beta)}{2\sigma^2}\Bigg) \end{equation}
that I obtained from assuming the natural conjugate prior for $\beta$ given $\sigma^2$: \begin{equation} p(\beta|\sigma^2) \propto |\sigma^2B|^{-1/2} \exp\Big(-\frac{1}{2}(\beta-b)'B^{-1}(\beta-b) \Big) \end{equation}
where $B$ and $\beta$ are prior parameters.
For $\sigma^2$ an uninformative prior is assumed: \begin{equation} p(\sigma^2) \propto \sigma^{-2} \end{equation}
Moreover can anyone recommend some resources (sites, textbooks) that allow one to practice such equations? I am quite comfortable with general matrix operations but struggle in equations like the one presented above.
$(Q^\frac12x)^T(Q^\frac12x) = x^T{Q^\frac12}^TQ^\frac12x = x^TQx$ always (when $Q$ has a Cholesky factor). And it is the same as $(-x)^TQ(-x)$ if that's what worries you.