TITLE: Transpose of block matrix
QUESTION [27 upvotes]: I'm attempting to prove that
$$
\left[ \begin{array}{c c}
A & B \\
C & D \\
\end{array} \right]^\top =
\left[ \begin{array}{c c}
A^\top & C^\top \\
B^\top & D^\top \\
\end{array} \right].
$$
Intuitively, I can see that it's true. However, when I try to formally prove it, I quickly get lost in the indices. What tricks can I use to keep things straight?
Source: Exercise 2.6.16, P116, Intro to Linear Algebra, 4th Ed by Strang

REPLY [41 votes]: Most people would just claim this is obvious and omit the proof, but if you don't want to do that then perhaps you could first prove that 
\begin{equation}
\begin{bmatrix} M & N \end{bmatrix}^T 
= \begin{bmatrix} M^T \\ N^T \end{bmatrix}
\end{equation}
and
\begin{equation}
\begin{bmatrix} M \\ N \end{bmatrix}^T 
= \begin{bmatrix} M^T & N^T \end{bmatrix}.
\end{equation}
Then
\begin{align}
\begin{bmatrix} A & B \\ C & D \end{bmatrix}^T 
&= \begin{bmatrix} 
\begin{bmatrix} A \\ C \end{bmatrix}^T \\
\begin{bmatrix} B \\ D \end{bmatrix}^T
\end{bmatrix} \\
&= \begin{bmatrix} A^T & C^T \\ B^T & D^T \end{bmatrix}.
\end{align}