Why can't we add a non-square matrix $A$ to its transpose $A^T$?

Question

The addition operation is commonly defined as follows:

Two matrices must have an equal number of rows and columns to be added

But this is a very shallow definition/interpretation.

A deeper interpretation of a $n \times n$ matrix, I would believe, would be that it is an element of a vector space, e.g. $A \in \mathbb{R}^{n \times n}$ as you need $n^2$ linear equations to compute each entry in the matrix (a dot product for each entry)

Now if you use that interpretation of a matrix, then let's assume we have a column vector $B \in \mathbb{R}^{n \times 1}$ and it's transpose, a row vector, $B^T \in \mathbb{R}^{1 \times n}$

$$\begin{align}B = \begin{bmatrix} b_1 \\ b_2\\ . \\ .\\ .\\ b_n\end{bmatrix} &&\text{and}&&B^T = \begin{bmatrix} b_1 & b_2& . & .& .& b_n\end{bmatrix} \end{align}$$

But both $B$ and $B^T$, are both elements of the same vector space $B, B^T \in \mathbb{R}^n$. So why is their addition undefined?

Why can't you add a matrix to its transpose, just like you could two vectors that are elements of the same vector space?

Is it wrong to interpret matrices as I have done? Is there a more rigorous definition of the addition operation for matrices? Are there better ways to interpret matrices?

Answer 1

$B$ and $B^T$ are not elements of the same vector space: $B$ is an element of $\Bbb R^{n\times 1}$, and $B^T$ is an element of $\Bbb R^{1\times n}$. Both of these vector spaces are isomorphic to $\Bbb R^n$, but no two of $\Bbb R^n$, $\Bbb R^{n\times 1}$, and $\Bbb R^{1\times n}$ are actually equal to each other.

Answer 2

I disagree that your proposed definition is "deeper". To me, it is more appropriate to think of $m\times n$ matrices at linear maps $\Bbb R^n\to\Bbb R^m$. From this perspective it is obvious why an $m\times n$ matrix can only be added to a $p\times q$ matrix when $m=p$ and $n=q$. Indeed, a linear map $\Bbb R^n\to\Bbb R^m$ can only be added to another linear map $\Bbb R^q\to\Bbb R^p$ if $n=q$ and $m=p$.

Answer 3

To add a layer of interpretation, an $n \times 1$ is often seen as a point in $\Bbb R^n$. A $1 \times n$ matrix will, in the same view, be a linear map $\Bbb R^n \to \Bbb R$. You may apply functions to points (by matrix multiplication), but you do not add functions and points.