The formula uses the Kronecker delta:
That is a short way of writing
But it doesn't make a whole lot of sense.
Yes, if "a" is just an nxn matrix then you have to find the two resulting matrices here: and then you solve the characteristic equation for this matrix ; i.e., to find its eigenvalues.
I would recommend looking at "easy" cases to start with (always a good strategy).
For n= 1, this operator is just "1+ a" and its operation is just multiplying the number x: for , of course.
For n= 2, this operator can be represented by the matrix . What are its eigenvalues?
For n= 3, this operator can be represented by the matrix .
What are its eigenvalues?