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Transpose a matrix whose elements are themselves matrices

by robotsheep
Tags: matrix, transpose
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robotsheep
#1
Nov21-12, 09:07 AM
P: 4
If I have (for simplicity) a vector ( A, B) where A and B are matrices how does the transpose of this look, is it ( AT, BT) or

(AT
BT)
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Robert1986
#2
Nov21-12, 10:05 AM
P: 828
Think about what the dimension should be.
robotsheep
#3
Nov21-12, 11:31 AM
P: 4
or

(A
B)

robotsheep
#4
Nov21-12, 11:44 AM
P: 4
Transpose a matrix whose elements are themselves matrices

Quote Quote by Robert1986 View Post
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
#5
Nov21-12, 01:54 PM
P: 15
From wikipedia:
In linear algebra, the transpose of a matrix A is another matrix AT (also written A′, Atr or At) created by any one 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
#6
Nov21-12, 06:01 PM
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Quote Quote by robotsheep View Post
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.

Quote Quote by robotsheep View Post
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 22 matrices for example, then I would interpret a notation like (A B) not as a 12 matrix whose elements are are 22 matrices, but as a 24 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 12 matrix whose elements are are 22 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 12 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
#7
Nov21-12, 06:14 PM
P: 4
Quote Quote by Fredrik View Post
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 22 matrices for example, then I would interpret a notation like (A B) not as a 12 matrix whose elements are are 22 matrices, but as a 24 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 12 matrix whose elements are are 22 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 12 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.


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