I'm learning about LU decomposition with a math book, but this question is not about LU decomposition, just wanted to explain why am I wondering about this.
We assume that $A$ is $m×n$ matrix, and $s = min(m,n)$ which compares two values and returns smaller value.
We can express $LU$ decomposition about $A$ with column/row block matrix like this:
$L = \begin{bmatrix}l_1 & \cdots & l_s\end{bmatrix}, U = \begin{bmatrix}u_1\\\vdots\\u_s\\\end{bmatrix}$
But the book expressed $LU$ decomposition to:
$L = \begin{bmatrix}l_1 & \cdots & l_s\end{bmatrix}, U = \begin{bmatrix}u_1^T\\\vdots\\u_s^T\\\end{bmatrix}$
Looks like each element of $U$ has been transposed. How could it be? Is it okay to do this?
I think entry $l_1,\ldots ,l_s$ and $u_1,\ldots ,u_s$ are themselves vectors, not numbers. To be specific, they are column vectors. Thus we line up column vectors a like $[l_1 l_2 \ldots l_s]$ we now get an $m\times s$ matrix, as each $l_i$ is a vector in dimension $m$. If we are stacking the $u_i$ (which are column vectors in dimension $n$) vertically we would get an very tall skinny $(sn)\times 1$ matrix. We don't want that, so we transpose them, turning each into row vectors and then stack them. Thus we get that $U$ is an $s\times n$ matrix.