# Matrix trace derivatives

em12
Hope this is the right section. I'm having trouble ironing out an apparent inconsistency in matrix trace derivative rules.

Two particular rules for matrix trace derivatives are

$$\frac{\partial}{\partial\mathbf{X}} Tr(\mathbf{X}^2\mathbf{A})=(\mathbf{X} \mathbf{A}+\mathbf{A} \mathbf{X})^T$$

and

$$\frac{\partial}{\partial\mathbf{X}} Tr(\mathbf{X}\mathbf{A}\mathbf{X}^T)=\mathbf{X} \mathbf{A}^T+\mathbf{X}\mathbf{A}$$

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

However, the first rule returns the derivative

$$- (\mathbf{X}\mathbf{A}+\mathbf{A}\mathbf{X})$$

and the second returns

$$2\mathbf{X}\mathbf{A}$$.

Am I missing something?