Derivative of a matrix with complex conjugates of multivariate functions [closed]

Question

I have a matrix:

U = {{0, f[x, t]}, {Conjugate[f[x, t]], 0}}

And I want to take its t-derivative to get:

{{0, f^(0,1)[x,t]},{Conjugate[f^(0,1)[x,t]], 0}}

Instead with the code from this question I get:

{{0, f^(0,1)[x,t]},{Conjugate`[f[x,t]]f^(0,1)[x,t], 0}}

How do I get the correct output?

Answer 1

There are System options one can set to control the behavior of D. One of these is the "ExcludedFunctions" option:

excluded="ExcludedFunctions"/.
    ("DifferentiationOptions"/.SystemOptions["DifferentiationOptions"])

{Hold,HoldComplete,Less,LessEqual,Greater,GreaterEqual,Inequality,Unequal,Nand,Nor,Xor,Not,Element,Exists,ForAll,Implies,Positive,Negative,NonPositive,NonNegative,Replace,ReplaceAll,ReplaceRepeated}

These are functions that D will not differentiate. We can add Conjugate to this list by using:

SetSystemOptions["DifferentiationOptions"->
    "ExcludedFunctions"->Union[excluded,{Conjugate}]]

DifferentiationOptions->{AlwaysThreadGradients->False,DifferentiateHeads->True,DifferentiateIteratorIndexed->True,DirectHighDerivatives->True,DirectHighDerivativeThreshold->10,ExcludedFunctions->{Conjugate,Element,Exists,ForAll,Greater,GreaterEqual,Hold,HoldComplete,Implies,Inequality,Less,LessEqual,Nand,Negative,NonNegative,NonPositive,Nor,Not,Positive,Replace,ReplaceAll,ReplaceRepeated,Unequal,Xor},ExitOnFailure->False,HighDerivativeMaxTerms->1000,SymbolicAutomaticDifferentiation->False}

Now, D will not try to differentiate Conjugate:

D[Conjugate[f[x]], x]//InputForm

Conjugate[Derivative[1][f][x]]

We are now free to give D rules for differentiating Conjugate:

Unprotect[Conjugate];
Conjugate /: D[Conjugate[f_], x__] := Conjugate[D[f, x]]
Protect[Conjugate];

Let's see what happens to the OP example now:

U = {{0, f[x, t]}, {Conjugate[f[x, t]], 0}};
D[U, t]

{{0,(f^(0,1))[x,t]},{Conjugate[(f^(0,1))[x,t]],0}}