How to divide a $1 \times n$ rowvector by a $n \times 1$ vector and the result of such a divsion I've been trying to search for this specific case, but can't seem to find the right answer which I guess is really simple.
Now I'm simply trying to find out how to do such a division and what is the end result, is it a scalar or a matrix of what dimensions?
We have the relation:
$Ao = \overrightarrow{v} * \bar{A}$
where $Ao$ is the orthogonal projections of the columns of matrix A onto the vector $\overrightarrow{v}$ and $\bar{A}$ is a rowvector containing the column averages of matrix A, and vector $\overrightarrow{v}$ consists of only $1$'s.
I want to find the matrix $\bar{A}$, so I believe I simply divide the vector $\overrightarrow{v}$ on both sides.
$\bar{A} = Ao/\overrightarrow{v}$
Do the inner dimensions have to agree? (i guess we can regard $\overrightarrow{v}$ as a matrix), they are as following:
$\bar{A} - 1 \times n$
$\overrightarrow{v} - n \times 1$

Any help is greatly appreciated.
 A: Dividing by a vector is something that is undefined. Let me give you some intuition about matrices to know why. Generally a matrix can be seen as an operator. Roughly speaking a matrix is something that acts on vectors and gives back an other vector.
You have row vector (aka 1xn matrix) and column vectors (aka nx1 matrix). If you use a row vector then you write $xA = y$, and then we say that matrix A manipulated row vector x to produce row vector y. If you use column vectors you write $Ax = y$, and we say that we manipulated x to produce vector y. Notice that with column vectors, we multiply from the right.
In most mathematical and engineering fields people use column vectors and we multiply with matrices to manipulate those vectors to change for instance the vectors x-direction or y-direction, etc. In other mathematical fields such as probability theory, people use row vectors and you multiply in $xA = y$ fashion.
Later I guess people generalized the definition of matrix-vector multiplication and added features such as matrix-matrix multiplication (which basically gives - in the case of multiplication of two matrices - as an output just the result that you would have had if you would have used the operator twice). The reason why you have so many plusses in your definition of matrix multiplication is because you want to go over all possible combinations (and take every output-to-input combination into account).
Even later people again generalized the notion and also allowed for individual rows and colums (just individual vectors) to be multiplied with each other. This had also interesting facts because it had nice applications such as the dot product of a row times a column.
In the story above what I wrote is probably historically incorrect, but the story makes sense and helps to understand the intuition behind these things.
To come back to your question. In my story there is only place for multiplications, since only they fit in my story of transforming vectors. Also know that matrices are very peculiar things and that a lot of things are not possible with matrices. For instance, matrices do not own the commutativity property in general, so even for 2 given nxn square matrices let us say A and B, $A * B \neq B * A$.
I hope this makes all a bit more sense now. If you have still questions, just ask!
Added later on due to an edit: The reason why matrix division does not exist is because the inverse of a matrix does not always exist. Not every matrix has an inverse $A^{-1}$. So you can not always divide! Does that make sense to you? When you divide you actually multiply by $A^{-1}$ but this is not always possible here. Some matrices however have inverses and these matriceses are called invertible or non-singular.
Added later on due to an edit: Is it this where you are talking about? (see at 9 minutes)
Good luck.
A: There is no such thing as a "vector division" that will give you what you want.  However, we can say that for any $i$ from $1$ to $n$: if $v_i$ is not zero, then
$$
\bar A = \frac 1{v_i} (A_{i1},A_{i2},\dots,A_{in})
$$
where $v_i$ denotes the $i$th entry of $v$ and $A_{ij}$ is the entry of $A$ in the $i$th row and $j$th column.
