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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.

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I think there are at least 2 questions here. Matrix acting on vectors is a linear operation. How do you define a non-linear matrix? With regards to hermitian operators, conjugate is usually defined as transposed and complex conjugate. Perhaps you could give an example you're working on? –  Valentin May 27 '12 at 22:13

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