I have $X$ a $1\times n$ row vector, and I need to convert it into a column vector through explicit matrix multiplications. That is equivalent to ask for the transpose of $X$, so in other words:
Is it possible to find a $A$ and $B$ so that
$AXB=X^{T}$
(Probably 2 matrices needed given the sizes). Do you have any idea how I can build $A$ and $B$? It should depend on $n$ of course, but I'd like to find a general expression.
Thanks a lot!
Note that if $X$ is $1 \times n$ and $AXB$ has size $n \times 1$, then $A$ must have size $n \times 1$ and $B$ must have size $n \times 1$. Thus, we have $$ AXB = A(XB) $$ where we note that $XB$ is $1 \times 1$, i.e. a scalar. Thus, we can only have $AXB = X^T$ if $X^T$ is a scalar multiple of $A$. Thus, there is no $A$ (and no combination of $A$ and $B$) that will satisfy $AXB = X^T$ for all $X$.