Say that one has a matrix representation of an operator A
with differential operators as entries in the matrix A
.
Is this a non-linear matrix? Since the differential is a linear operator and A
is composed of linear operators, I'm leaning towards A
being a linear operator.
If one were to take the conjugate transpose of A
, would the differential operators be modified? I'm trying to prove that A
is anti-hermitian, and it seems to me that the differential operators would have to be negated when A
is conjugate-transposed in order for A
to be anti-hermitian.