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Transpose a matrix whose elements are themselves matrices

  1. Nov 21, 2012 #1
    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)
     
  2. jcsd
  3. Nov 21, 2012 #2
    Think about what the dimension should be.
     
  4. Nov 21, 2012 #3
    or

    (A
    B)
     
  5. Nov 21, 2012 #4
    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.
     
  6. Nov 21, 2012 #5
    From wikipedia:
    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)
     
  7. Nov 21, 2012 #6

    Fredrik

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    Gold Member

    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}##.
     
    Last edited: Nov 21, 2012
  8. Nov 21, 2012 #7
    Thank you, this really cleared it up for me.
     
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