Confused about diagonal matix notation

Question

Reading a book of physics I found the following definition of diagonal matrix:

$$A_{ij}= A_{ii}\delta_{ij}$$

I understand a diagonal matrix has only diagonal elements nonzero, but is the previous notation correct?

I'm somehow confused because if we choose $A_{ij}$ with $i\neq j $ (f.e $A_{12}$) then we need to use $A_{11}$ from the right side, which seems a bit forced.

Answer 1

This notation uses the Kronecker delta $\delta_{ij}$, which is $1$ when $i=j$ and $0$ otherwise $($so, $0$ outside the diagonal$)$.

Albeit correct, I must say this is a very lazy definition of diagonal matrix.

Answer 2

It's correct, if $i=j$, $A_{ij}=A_{ii}\delta_{ij}=A_{ii}\cdot 1 = A_{ii}$

if $i\ne j$, $A_{ij}=A_{ii}\delta_{ij}=A_{ii}\cdot 0 = 0$