# Derivative of matrix involving trace and log

I'm stuck on this problem. Let $X\in\mathbb{R}^{n\times n}$, compute the following matrix derivatives $$\frac{\partial}{\partial X}\mathrm{tr}(\log(XA)\log(XA)^\top),$$ $$\frac{\partial}{\partial X}\mathrm{tr}(B\log(XA)),$$ where $\log(\cdot)$ is the matrix logarithm (not element-wise) and $A,B\in\mathbb{R}^{n\times n}$ are constant matrices.

Thanks in advance for your suggestions!

• Do you know the chain rule and product rule? – wj32 Jul 4 '13 at 22:34
• Of course in the scalar case, but I still have a doubt: are these rules applicable without changes also in the matrix case? How about the order of derivation in the latter case? Thanks – Ludwig Jul 5 '13 at 6:23

## 3 Answers

I assume you are able to compute the derivative without trace. And the rest part, actually, is not hard. Try to compute it componentwisely, then \begin{align} \frac{\partial}{\partial X^i_j}\text{tr}f(X)=& \frac{\partial}{\partial X^i_j}f^k_l(X)\delta^l_k\\ =&\frac d{dx}f^k_m(x)|_{x=X}\frac{\partial}{\partial X^i_j}X^m_l\delta^l_k\\ =&\frac d{dx}f^k_m(x)|_{x=X}\delta^m_i\delta^j_l\delta^l_k\\ =&\frac d{dx}f^j_i(x)|_{x=X} \end{align} The conclusion is

$$\frac{\partial}{\partial X}\text{tr}f(X)=\left(\frac d{dx}f(x)|_{x=X}\right)^T$$

Clearly the calculation is quasi-unfeasible. In particular, the answer of GMB is absolutely false (cf. my comment above). Who has given a good point to his answer ? Yet, one can give a compact answer if $X=A^{-1}$ because $D\log_I=id$.

Let $f(X)=tr(B\log(XA))$. Then $Df_{A^{-1}}:H\rightarrow tr(BHA)$. Thus $\nabla (f)_{A^{-1}}=(AB)^T$.

EDIT: Let $h(X)=tr(\log(I+X))$ where $\log$ denotes the principal logarithm and $||X||<1$. Then $h(X)=tr(\sum_{k=1}^{\infty}(-1)^{k+1}/k.X^k) =\sum_{k=1}^{\infty}(-1)^{k+1}/k.tr(X^k)$ and $Dh_X:H\rightarrow \sum_{k=1}^{\infty}(-1)^{k+1}/k.tr(kX^{k-1}H)$ (because $tr(UV)=tr(VU)$)$=tr(\sum_{k=1}^{\infty}(-1)^{k-1}X^{k-1}H)=tr((I+X)^{-1}H)$. In other words $\nabla(h)_X={(I+X)^{-1}}^T$. More generally, if $X$ has no eigenvalues in $\mathbb{R}^-$, then let $g(X)=tr(\log(X))$. One has $Dg_X:H\rightarrow tr(X^{-1}H)$ and $\nabla(g)_X={X^{-1}}^T$.

Here $f(X)=tr(B\log(XA))=tr(B\log(Y))$; then we must derive $u(Y)=tr(BY^k)$. Unfortunately $Du_Y:K\rightarrow tr(B(kY^{k-1})K)$ is false in general ; yet, it is true if, for instance, $BY=YB$. Finally, when $BXA=XAB$, $Df_X:H\rightarrow tr(B(XA)^{-1}HA)$ (because $K=HA$)$=tr((XA)^{-1}BHA)=tr(X^{-1}BH)$ and $\nabla(f)_X=(X^{-1}B)^T$ does not depend on $A$.

Here's an incomplete answer:

$\log A = - \sum_{i = 1}^{\infty} \, \frac{1}{i}(I - A)^i$

So

$d \log A = \sum_{i = 0}^{\infty} \, (I - A)^i \, dA$

If you know the chain and product rules, deriving over the $\log A$ term should be the only tricky part.

• Thanks for your answer. However, I was looking for a closed-form expression (if there exists). – Ludwig Jul 5 '13 at 6:27
• Hi Levdub,the notation $df$ is really catastrophic. Here $A$ and $dA$ do not commute. The derivative of the function $log$ (and of the function $exp$) is complicate. For instance, the derivative of $f(A)=-(I-A)^3$ is $Df_A:H\rightarrow H(I-A)^2+(I-A)H(I-A)+(I-A)^2H$. – loup blanc Sep 5 '13 at 12:47