- #1

em12

- 3

- 0

Two particular rules for matrix trace derivatives are

[tex]\frac{\partial}{\partial\mathbf{X}} Tr(\mathbf{X}^2\mathbf{A})=(\mathbf{X} \mathbf{A}+\mathbf{A} \mathbf{X})^T[/tex]

and

[tex] \frac{\partial}{\partial\mathbf{X}} Tr(\mathbf{X}\mathbf{A}\mathbf{X}^T)=\mathbf{X} \mathbf{A}^T+\mathbf{X}\mathbf{A}[/tex]

Now assume that [tex]\mathbf{A}[/tex] is diagonal (or maybe even just symmetric) and [tex] \mathbf{X}[/tex] is anti-symmetric. Then by the cyclic property of the trace, [tex]-Tr(\mathbf{X}^2\mathbf{A})=Tr(\mathbf{X}\mathbf{A}\mathbf{X}^T)[/tex]. So the two derivatives should be equal up to a minus sign, no?

However, the first rule returns the derivative

[tex]- (\mathbf{X}\mathbf{A}+\mathbf{A}\mathbf{X})[/tex]

and the second returns

[tex] 2\mathbf{X}\mathbf{A}[/tex].

Am I missing something?