試題詳解

10. Let A be an n x n matrix whose (i, j) component is aij. Let f be a real-valued function defined on A. Let  ∇_Af( A ) be the gradient of f( A ) with respect to A； ∇_Af( A )is delined as an n x n matrix whose (i, j) entry is . Let B and C be n x n matrices. Which of the following statements are true?
(A) If f( A ) = tr(AB), then ∇_Af( A )= B.
(B) If f( A ) = tr(AB), then ∇_Af( A ) = B^T.
(C) If f( A ) = tr(AA^TC), then ∇_Af( A ) = CA + C^TA^T.
(D) If f( A ) == tr(AA^TC), then ∇_Af(  A  ) = CA + CA^T.
(E) None of the above.