Equations with alignment and with centering between alignment

Question

I have the following code:

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\begin{align}
&\begin{bmatrix}
a'_{1,1} & a'_{1,2}\\
a'_{2,1} & a'_{2,2}
\end{bmatrix} &&= 
\begin{bmatrix}
\cos{\alpha} & \sin{\alpha} \\
-\sin{\alpha} & \cos{\alpha} \\
\end{bmatrix}
&&\cdot
&&\begin{bmatrix}
a_{1,1} & a_{1,2}\\
a_{2,1} & a_{2,2}
\end{bmatrix}
&&\cdot
&&\begin{bmatrix}
\cos{\alpha} & -\sin{\alpha} \\
\sin{\alpha} & \cos{\alpha} \\
\end{bmatrix}
\\
&\textbf{A}' &&= \textbf{R}^T &&\cdot && \textbf{A} &&\cdot &&\textbf{R}
\end{align}
\end{document}

It aligns indeed all the arguments like this:

But I want to have all the arguments centered between the separation as follows and without the equation numbering:

Answer 1

A seven column array does the job. The columns for = and \cdot are of type >{{}}c<{{}} so the spacing around the items is the expected one. The outer array has no intercolumn spaces.

\documentclass{article}
\usepackage{amsmath,array}

\begin{document}

\begin{equation*}
\begin{array}{ @{}c *{3} { >{{}}c<{{}} @{} c @{} } }
\begin{bmatrix}
a'_{1,1} & a'_{1,2}\\
a'_{2,1} & a'_{2,2}
\end{bmatrix}
&=&
\begin{bmatrix}
\cos{\alpha} & \sin{\alpha} \\
-\sin{\alpha} & \cos{\alpha} \\
\end{bmatrix}
&\cdot&
\begin{bmatrix}
a_{1,1} & a_{1,2}\\
a_{2,1} & a_{2,2}
\end{bmatrix}
&\cdot&
\begin{bmatrix}
\cos{\alpha} & -\sin{\alpha} \\
\sin{\alpha} & \cos{\alpha} \\
\end{bmatrix}
\\[2ex]
\textbf{A}' &=& \textbf{R}^T &\cdot& \textbf{A} &\cdot &\textbf{R}
\end{array}
\end{equation*}

\end{document}

Answer 2

Here's a suggestion: Take a different approach to denoting what's A', A, R, R^T, by using \underbrace directives. This should make it clear to even the most casual readers that they're not dealing with two separate equations.

\documentclass{article}
\usepackage{mathtools} % for 'bmatrix*' env.
\begin{document}
\begin{equation}
{\underbrace{\begin{bmatrix}
  a'_{1,1} & a'_{1,2}\\
  a'_{2,1} & a'_{2,2}
\end{bmatrix}}_{\textstyle\mathbf{A}'}} = 
{\underbrace{\begin{bmatrix*}[r]
   \cos{\alpha} & \sin{\alpha} \\
  -\sin{\alpha} & \cos{\alpha} \\
\end{bmatrix*}}_{\textstyle\mathbf{R}^T}}
\cdot
{\underbrace{\begin{bmatrix}
  a_{1,1} & a_{1,2}\\
  a_{2,1} & a_{2,2}
\end{bmatrix}}_{\textstyle\mathbf{A}}}
\cdot
{\underbrace{\begin{bmatrix*}[r]
  \cos{\alpha} & -\sin{\alpha} \\
  \sin{\alpha} &  \cos{\alpha} \\
\end{bmatrix*}}_{\textstyle\mathbf{R}}}
\end{equation}
\end{document}