# Chain rule on matrix differentiation of trace in Kalman gain proof

I was reading the following derivation of the Kalman filter gain http://www.robots.ox.ac.uk/~ian/Teaching/Estimation/LectureNotes2.pdf on pg 5-6.

From the following

$$L=\min_{K_{k+1}} trace(P_{k+1|k+1})$$

With

$$P_{k+1|k+1} = \boldsymbol{ (I-K_{k+1} H_{k+1}) P_{k+1|k} (I-K_{k+1} H_{k+1})^{T} + K_{k+1}R_{k+1}K_{k+1}^T }$$

K,H, and P are all nxn matrices.

It uses the following trace differentiation identity:

$${\frac{\partial}{\partial \boldsymbol{A}}} (trace(\boldsymbol{ABA^T})) = 2\boldsymbol{AB}$$

But from there, there is a bit of a skip to the result of the differentiation

$${\frac{\partial L}{\partial \boldsymbol{ K_{k+1}}}} = -2(\boldsymbol{I-K_{k+1}H_{k+1})P_{k+1|k}H_{k+1}^T+2K_{k+1}R_{k+1} } = 0$$

What I am wondering about is the origin of the $H_{k+1}^T$ term in the above equation.

It appears to me that chain rule is applied here. But most online sources on matrix chain rule or trace identities does not show a direct example that the chain rule can be applied to the derivative of a trace.

ie. This is what I think happens in the intermediate step:

$$\boldsymbol{A=I-K_{k+1} H_{k+1}; B=P_{k+1|k}}$$

$${\frac{\partial}{\partial \boldsymbol{K_{k+1}}}} (trace(\boldsymbol{ABA^T})) = 2\boldsymbol{AB} * {\frac{\partial (trace (\boldsymbol{A}))}{\partial \boldsymbol{ K_{k+1}}}}$$

Question: Is the above chain rule application valid? If so, I would appreciate a pointer to a proof.

And then using

$${\frac{\partial trace(\boldsymbol{K_{k+1}H_{k+1}} )} {\boldsymbol{\partial K_{k+1}}}} = \boldsymbol{H_{k+1}^T}$$

To arrive at

$$= -2(\boldsymbol{I-K_{k+1} H_{k+1}) * P_{k+1|k} * H_{k+1}^T}$$

If the intermediate steps is actually different, please let me know and link to any proofs of the identities used as I am self-learning linear algebra. Thanks.

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## 1 Answer

Firstly $\nabla_A(ABA^T)=A(B+B^T)=2AB$ only if $B$ is symmetric.

Here $L=trace((I-KH)Q(I-KH)^T+KRK^T)$ where $Q,R$ are symmetric matrices. The derivative is $DL_K:Z\rightarrow trace(-ZHQ(I-KH)^T+(I-KH)Q(-ZH)^T+ZRK^T+KRZ^T)=trace(-2ZHQ(I-KH)^T+2ZRK^T)=trace((-2HQ(I-KH)^T+2RK^T)Z)$.

I wrote that follows in at least 10 posts but nobody reads my post (tears). Yet, no students know how to calculate a gradient. Incredible !

A gradient is associated to a scalar product; we take, here, $(U,V)=trace(U^TV)$. The associated gradient is defined by, for every $Z$, $(\nabla_K(L),Z)=DL_K(Z)$. Thus $\nabla_K(L)=-2(I-KH)QH^T+2KR$.

EDIT (answers to frank): 1),2) $DL_K$ is a linear application and $Z$ is the variable. For instance, if $f:K\rightarrow KAK$, then $Df_K:Z\rightarrow ZAK+KAZ$, the derivative of a product ! ($(uv)'=u'v+uv'$). If $f:K\rightarrow -KHQ$, then $Df_K:Z\rightarrow -ZHQ$. $(ua)'=u'a$.

3) Here $Df_X:(h,k,l)\rightarrow [3z,4y^3,3x][h,k,l]^T$. We use the scalar product over vectors $(u,v)=trace(u^Tv)=u^Tv$. Then $\nabla_Xf=[3z,4y^3,3x]^T$. More generally, to define a gradient with the help of a scalar product has a geometric meaning. You do not need a basis.

4) The derivative uses trace and the gradient not. It is linked to the chosen scalar product.

5) Consider the application $f:K\in\mathcal{M}_n(\mathbb{R})\rightarrow KAK\in\mathcal{M}_n(\mathbb{R})$, then the derivative is the linear application $Df_K:Z\in\mathcal{M}_n(\mathbb{R})\rightarrow ZAK+KAZ\in\mathcal{M}_n(\mathbb{R})$. In the same way, if $g(K)=K^3$, then $Dg_K(Z)=ZK^2+KZK+K^2Z$. If you choose coordinates, then the matrix of the derivative is its Jacobian. For 3), $f:\mathbb{R^3}\rightarrow \mathbb{R}$ and let $X=[x,y,z]^T$. $Df_X$ is a linear application $(h,k,l)\in\mathbb{R^3}\rightarrow \mathbb{R}$ and , consequently, its matrix (the Jacobian) is $1\times 3$, that is a row. The gradient is the transpose of the previous matrix, that is a vector.

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Thanks, a few follow-ups – frank Oct 16 '13 at 4:08
1) What is variable Z in the first part $DL_K:Z$? $DL_K$ is already taking the derivative of L with respect to K. And is the : Z the Frobenius inner product notation? Why is that notation necessary? 2) Since trace is linear, d tr(X) = tr(dX). When you are taking the derivative of the trace, how does the -KH)Q term become -ZHQ? Seems like the product rule, for the first term, you held (I-KH)^T constant. But K, H, Q are nxn, and d(-KHQ)/dK would be the derivative of a matrix by a matrix, which should be undefined? – frank Oct 16 '13 at 4:21
3) I know how to calculate a gradient like so from math.umn.edu/~mathe233/math2263/resources/… If f(x,y,z) = 3xz + y^4 Df(x,y,z) = [3z 4y^3 3x] $\nabla_f(x,y,z) = (3z, 4y^3, 3x)$ Using the expression above, Our K is like the x,y,z. the f function is like the L. So the expression of taking the gradient of L with respect to K I understand, but the additional taking the Frobenius product with Z I don’t understand. – frank Oct 16 '13 at 4:26
4) I read (U,V) is a notation for Frobenius inner product. $DL_K:Z$ is also a Frobenius inner product notation, so $DL_K:Z$ from your reply is the same as ($DL_K$, Z) = trace(U^TV). And we have V=Z, we get $DL_K$ which is what we wanted to solve. Why do we need the extra expression for ($\nabla_K(L),Z) =$DL_K$(Z)? Gradient and matrix derivative are largely same, no? pg 4 of math.umn.edu/~mathe233/math2263/resources/… – frank Oct 16 '13 at 4:28 5) Thanks for the update. I think I get the idea of the steps, so I will accept the answer. In 1) the switch to Z still looks funny, "if$f:K\rightarrow KAK$, then$Df_K:Z\rightarrow ZAK+KAZ$". The K in$Df_K$should already specify the derivative is with respect to K, is there a difference between saying$Df_K:Z$and$Df_K:K$? What is Z equal to? I tried to search for "linear application" to see if I can find more examples, but that combination in web search turn out too many random things. In 3), I did not have h,k,l, but you had$Df_x : (h,k,l), is that same as Df_x : (x,y,z)? – frank Oct 17 '13 at 7:31