矩陣求導

Matrix calculus

1. Derivative of linear function:

   ∂xTa∂x=a

   ∂aTx∂x=a

   ∂xTXb∂X=abT

   ∂xTXTb∂X=baT

2. Derivative of quadratic function:

   ∂bTXTXc∂X=X(bcT+cbT)

   ∂(Bx+b)TC(Dx+d)∂x=BTC(Dx+d)+DTCT(Bx+b)

   ∂xTBx∂x=(B+BT)x

   ∂bTXTDXc∂X=DTXbcT+DXcbT

3. Derivative of determinant:

   ∂det[Y]∂x=det[T]tr[Y−1∂Y∂x]

   ∂det[Y]∂Y=det[Y]Y−T

4. Derivative of log determinant:

   ∂log[det[Y]]∂Y=Y−T

5. Derivative of inverse:

   ∂Y−1∂x=−Y−1∂Y∂xY−1

6. Derivative of trace:

   ∂tr[F[X]]∂X=(∂F[X]∂X)T