In this video on Tensor calculus(timestamped), the professor explains how the matrix entry is written in a tensor-like notation:
$$\text{(i,j)}^{th}\text{ entry of matrix A}=A_{ij}=A_j^i$$
Later in the video, he also proceeds to explain why $A_i^i$ represents the trace of the matrix, which is a natural consequence of the Einstein summation notation.
The question here is, how would you differentiate the k$^{th}$ diagonal entry of the matrix, $A_{kk}$, from it being a live index hence representing a sum?
In practice, one never refers to a diagonal element of a matrix because it is something that depends on the coordinate system. Einstein's notation only lets you write complete expressions that denote things that do not depend on that.
If you for some reason need to write something that does not satisfy that condition it is useful to be explicit independently of what notation you are using!