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robotsheep
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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)
(AT
BT)
Robert1986 said:Think about what the dimension should be.
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
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.robotsheep said:Sorry, I don't really understand what you mean by "dimension" in this case;
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) isrobotsheep said: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.
Fredrik said: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}##.
A matrix is a mathematical concept used to organize and manipulate data. It is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are often used in linear algebra and are useful in solving equations, representing transformations, and conducting statistical analyses.
Transposing a matrix means to switch its rows and columns. This results in a new matrix with the rows of the original becoming the columns of the new matrix and vice versa. The dimensions of the transposed matrix will be the reverse of the original, meaning if the original matrix was m x n, the transposed matrix will be n x m.
To transpose a matrix whose elements are themselves matrices, you must first transpose each individual element. Once the elements have been transposed, you can then transpose the entire matrix like you would a regular matrix. This will result in a matrix of matrices with the rows and columns switched.
Transposing a matrix of matrices can be useful in certain mathematical operations, such as matrix multiplication. It can also help in organizing and analyzing data when the elements of the matrix represent different variables or categories. Additionally, transposing can help simplify and clarify complex data sets.
Transposing a matrix of matrices can be used in various fields, including statistics, computer science, and economics. In statistics, it can be used in multivariate analyses and data visualization. In computer science, it can be used in image processing and machine learning. In economics, it can be used in input-output analysis and production theory.