# Order of transposes for matrix with matrix elements?

So say I have a matrix $$\begin{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix}^T & \begin{bmatrix} 3 \\ 4 \end{bmatrix}^T & \begin{bmatrix} 5 \\ 6 \end{bmatrix}^T \end{bmatrix}^T.$$

How would I simplify this? I've tried two ways and they give me different resultant matrices, so I wanted to know if there's any specific order on these. For the example above, do we treat each inner matrix as a single element of the outer one, or do we treat all the inner matrices' elements as elements of the outer one?

I'm asking mostly because I'm trying to understand the construction of a certain matrix in a paper I'm reading, and I'm not fully sure if I've seen something like $$A = \begin{bmatrix} a^{(1)^T} & \dots & a^{(n)^T} \end{bmatrix}^T$$ before, (where the $$a^{(i)}$$'s are matrices as well).

It would be super helpful if anyone could explain. Thank you!

• Is the paper publicly available? Perhaps it would be possible to infer which of the two meanings they have in mind from the context?
– user700480
Commented Sep 6, 2022 at 10:26

The usual interpretation of a notation like this is that $$\begin{bmatrix} A^T & B^T & C^T \end{bmatrix}$$ is a matrix in block format, so it is really just a single matrix composed of the elements that belonged to $$A^T,$$ $$B^T,$$ and $$C^T.$$ When you transpose it, its first row becomes its first column, etc., so
$$\begin{bmatrix} A^T & B^T & C^T \end{bmatrix}^T = \begin{bmatrix} A \\ B \\ C \end{bmatrix}.$$
For example, $$\begin{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix}^T & \begin{bmatrix} 3 \\ 4 \end{bmatrix}^T & \begin{bmatrix} 5 \\ 6 \end{bmatrix}^T \end{bmatrix}^T = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \end{bmatrix}^T = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \end{bmatrix}$$
A reason to use the transpose within the transpose is that you really wanted the blocks of the matrix to be arranged in a column like this, $$\begin{bmatrix} A \\ B \\ C \end{bmatrix},$$ but you want to write the matrix as a formula inline with your text, or you just don't want to use that much vertical space.