I've never really worked with matrices so I would be glad about some information on the following issue:
In some statistic calculations there is often a transposed matrix within a formula. Can someone explain why there is a need to transpose matrices and what properties do they have afterwards?
Here is an example of the Shapiro-Wilk test for normality:
$a_{i} = \left[ (m^T V^{-1} V^{-1} m)^{\frac{1}{2}} \right] m^T V^{-1}$
where $V = \left( \begin{array}{*{3}, c} cov(m_1, m_1) & \cdots & cov(m_1, m_n) \\ \vdots & \ddots & \vdots \\ cov(m_n, m_1) & \cdots & cov(m_n, m_n) \end{array} \right)$
and $m_i = \Phi^{-1} \left( \frac{ i - \frac{3}{8}}{ n + \frac{1}{4}} \right)$
Why is $m_i$ a matrix and why should it be transposed for calculation?