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Am learning about Fréchet differentials and was wondering if for a real matrix $X$ and positive semidefinite real matrices $A,B$ the function $f(X)=TrX^TAX-X^TBX$ is twice Fréchet differentiable or not?

I believe that the gradient is $2AX-2BX$ but how do I prove Fréchet differentiability?

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    If all 1st order partial derivatives are continuous, the function is Fréchet differentiable.2012-09-14

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