Doubt about transpose operation in SVD matrix tutorial

Question

I am trying to get a grasp of the Singular value Decomposition of a matrix following this tutorial.

I am confused however by the way they transpose the matrix A. From what I understood from the transposition operation,
|A |B | ----------------------------- |A |C |0 |0 |
|C |D | --- should transpose to --- |B |D |0 |0 |
|0 |0 |
|0 |0 |

Or did I misunderstand the transpose process?

Answer 1

Your understanding is correct; that's how transposes work.

I think you're asking because the page shows the computation of $A A^T = W$, but the transpose $A^T$ is incorrect. Their given value of $W$ is correct according to the conventional definition of transpose (which you gave), so it seems that it's only $A^T$ that's incorrectly written.

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Given that \begin{equation} A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} \end{equation} the transpose should be \begin{equation} A^T = \begin{bmatrix} 2 & 1 & 0 & 0 \\ 4 & 3 & 0 & 0 \end{bmatrix} \end{equation} and not \begin{equation} A^T \color{red}{\neq} \begin{bmatrix} 2 & \color{red}{4} & 0 & 0 \\ \color{red}{1} & 3 & 0 & 0 \end{bmatrix} \end{equation} which your linked tutorial gave.

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It might be nice to email the owner of the page to clue them in. :)