Sometimes your notation is used. The problem of the ambiguity of $X_{m \times n}^T$ comes from the wrong usage. There is no such thing as $X_{m \times n}^T$. Instead, there is $X_{m \times n}$ and there is $X^T$. Combining these two will give you either $(X_{m \times n})^T$ or $(X^T)_{m \times n}$, and removing the parentheses implies that $*_{* \times *}$ and $*^T$ commute (like, for example, $X^{-T} = (X^{-1})^T = (X^T)^{-1}$), which is wrong.
However, I see very little, if any, practical usage in this kind of typesetting. More or less standard way is $X \in \mathbb{R}^{m \times n}$ or $X \in M_{m \times n}(\mathbb{R})$, and then you just use $X$. Writing dimensions in the formulas might make sense at the very beginning of learning this stuff, but not for long, and in this case, parentheses also make a lot of sense.
For anything more advanced, let me give you an example: would you, in a similar fashion, write
$$x_\text{even} + y_\text{odd} = z_\text{odd}?$$
Properties of the objects usually unnecessarily clutter your formulas, and are far better to be declared before the first use, instead of all the time. IMO, of course.
Well, formally a matrix of type $(I, J)$ is just a function from $I × J$. $I$ is an index set for rows and $J$ for columns. Usually $I = \{1, …, m\}$, $J = \{1, …, n\}$ for some $m, n ∈ \mathbb{N}$, but nothing stops you to have $I = \{A, B, C, D, E\}$, $J = \{A, B, C\}$. Since a matrix, say $X$, is just a function, its element in the $i$-th row and $j$-th column is just $X(i, j)$. Traditionally $x_{ij}$ is used instead of $X(i, j)$ but the semantics is the same. Now we have $X = (x_{ij})$ where $(x_{ij})$ is really a shortcut for $(x_{ij}: i ∈ I, j ∈ J)$ which is function analogue of set-builder notation $\{x_{ij}: i ∈ I, j ∈ J\}$.
If you want the sum of $i$-th row then it is in general $\sum_{j ∈ J} x_{ij}$ which is variant of $\sum (x_{ij}: j ∈ J)$ and depends on $i$. In case that $J = \{1, …, n\}$, then $\sum_{j = 1}^n x_{ij}$ is used as a shortcut for $\sum_{j ∈ \{1, …, n\}} x_{ij}$. Note the differences to your notation: there are no parentheses since $x_{ij}$ in the sum is just a particular element rather then the matrix $(x_{ij})$. And other thing is that first index is traditionally row index and second is column index so in your example you are actually summing over elements of $j$-th row.
Also, if the index set you are summing over is clear from context then you can write just $\sum_j x_{ij}$. With this shortcut it is easy to write your value as $\sum_j x_{ij} - x_{ii} = \sum_{j ≠ i} x_{ij}$.
Also note that the limits are typed to the right from the sum sign rather then above and below just because its inside a paragraph. In display mode it looks like
$$
\sum_{j = 1}^n x_{ij},
$$
but the meaning is the same.
Best Answer
Simply put, $\Phi=(1,1,\ldots,1)\phi$.