Transpose a matrix whose elements are themselves matrices

[robotsheep]

If I have (for simplicity) a vector ( A, B) where A and B are matrices how does the transpose of this look, is it ( A^T, B^T) or

(A^T
B^T)

 

[Robert1986]

Think about what the dimension should be.

 

[robotsheep]

or

(A
B)

 

[robotsheep]

Think about what the dimension should be.

Sorry, I don't really understand what you mean by "dimension" in this case;

I know that the transpose of a 1x2 matrix should be a 2x1 matrix but I don't know whether the elements actually inside the matrix should be transposed once I make the matrix a 2x1.

Thank you in advance for any help.

 

[aija]

From wikipedia:

In linear algebra, the transpose of a matrix A is another matrix AT (also written A′, Atr or At) created by anyone of the following equivalent actions:
reflect A over its main diagonal (which runs top-left to bottom-right) to obtain AT
write the rows of A as the columns of AT
write the columns of A as the rows of AT

It doesn't say that anything should be done to the elements of the matrix so I guess it would be just
A
B
(columns of A written as rows)

 

[Fredrik]

Sorry, I don't really understand what you mean by "dimension" in this case;

He meant that you should think about the number of rows and columns. However, this only helps is you interpret the notation in the first of the two ways I'm describing below.

I know that the transpose of a 1x2 matrix should be a 2x1 matrix but I don't know whether the elements actually inside the matrix should be transposed once I make the matrix a 2x1.

If A and B are 2×2 matrices for example, then I would interpret a notation like (A B) not as a 1×2 matrix whose elements are are 2×2 matrices, but as a 2×4 matrix whose 11, 12, 21 and 22 elements are respectively the 11, 12, 21, 22 elements of A, and whose 13, 14, 23, 24 elements are respectively the 11, 12, 21, 22 elements of B. With this interpretation of the notation, it's obvious that the transpose of (A B) is
$$\begin{pmatrix}A^T\\ B^T\end{pmatrix}.$$ If you instead interpret it as a 1×2 matrix whose elements are are 2×2 matrices, then the standard definition of "transpose" would of course just give you
$$\begin{pmatrix}A\\ B\end{pmatrix}.$$ I think the former interpretation is far more useful, and I assume that to some authors, this is a reason to use a different definition of "transpose", so that you can think of (A B) as a 1×2 matrix, and still have its transpose be
$$\begin{pmatrix}A^T\\ B^T\end{pmatrix}.$$ I don't see any reason to use a definition that makes $\begin{pmatrix}A & B\end{pmatrix}^T=\begin{pmatrix}A^T & B^T\end{pmatrix}$.

 

[robotsheep]

He meant that you should think about the number of rows and columns. However, this only helps is you interpret the notation in the first of the two ways I'm describing below.


If A and B are 2×2 matrices for example, then I would interpret a notation like (A B) not as a 1×2 matrix whose elements are are 2×2 matrices, but as a 2×4 matrix whose 11, 12, 21 and 22 elements are respectively the 11, 12, 21, 22 elements of A, and whose 13, 14, 23, 24 elements are respectively the 11, 12, 21, 22 elements of B. With this interpretation of the notation, it's obvious that the transpose of (A B) is
$$\begin{pmatrix}A^T\\ B^T\end{pmatrix}.$$ If you instead interpret it as a 1×2 matrix whose elements are are 2×2 matrices, then the standard definition of "transpose" would of course just give you
$$\begin{pmatrix}A\\ B\end{pmatrix}.$$ I think the former interpretation is far more useful, and I assume that to some authors, this is a reason to use a different definition of "transpose", so that you can think of (A B) as a 1×2 matrix, and still have its transpose be
$$\begin{pmatrix}A^T\\ B^T\end{pmatrix}.$$ I don't see any reason to use a definition that makes $\begin{pmatrix}A & B\end{pmatrix}^T=\begin{pmatrix}A^T & B^T\end{pmatrix}$.

Thank you, this really cleared it up for me.