I am trying to prove the relationship (4.65) expressed in Bishop's book (Pattern Recognition and Machine Learning) in chapter (4.2.1) Continuous inputs:
I found the following passages but there is one passage that I cannot understand the author's motivation:
I report the unclear passage:
$\frac{1}{2} \left [ \left ( \vec{x} - \vec{\mu_{1}} \right )^{T} \left ( \Sigma^{-1} \vec{x} - \Sigma^{-1} \vec{\mu_{1}} \right ) \right ]$
for me:
$ \frac{1}{2} \left [\vec{x}^{T}\Sigma^{-1} \vec{x} - \vec{x}^{T}\Sigma^{-1} \vec{\mu_{1}} - \vec{\mu_{1}}^{T}\Sigma^{-1} \vec{x} + \vec{\mu_{1}}^{T}\Sigma^{-1} \vec{\mu_{1}}\right ]$
but the author considers the following two terms to be equal:
$- \vec{x}^{T}\Sigma^{-1} \vec{\mu_{1}}$ and $-\vec{\mu_{1}}^{T}\Sigma^{-1} \vec{x} $
If $u,v$ are $n\times 1$ matrices and $A$ is an $n\times n$ matrix, then as stated in the comments, $$(u^TAv)^T=u^TAv.$$ On the other hand, by properties of the transpose, $$(u^TAv)^T=v^TA^Tu.$$ If $A=A^T$, then $$u^TAv=(u^TAv)^T=v^TA^Tu=v^TAu.$$