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Assume $S_1$ and $S_2$ are two $n \times n$ (positive definite if that helps) matrices, $c_1$ and $c_2$ are two variables taking scalar form, and $u_1$ and $u_2$ are two $n \times 1$ vectors. In addition, $c_1+c_2=1$, but in the more general case of $m$ $S$'s, $u$'s, and $c$'s, the $c$'s also sum to 1.

What is the derivative of $(c_1 S_1+c_2 S_2)^{-1}(c_1 u_1+c_2 u_2)$ with respect to both $c_1$ and $c_2$?

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    Have you tried using the formula for the derivative of the inverse of a matrix? http://en.wikipedia.org/wiki/Invertible_matrix#Derivative_of_the_matrix_inverse2011-01-14
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    I am familiar with that formula. I am not sure how to proceed afterward. I do know that if S=S1=S2, then the answer should be S^-1*u1 for c1 and S^-1*u2 for c2. When I tried to apply the inverse rule for the more general problem, I could not find a similar result.2011-01-14
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    @John: If that's what you get when $S=S_1=S_2$, then perhaps I misunderstood your problem. The way I am interpreting it, you cannot replace $c_1+c_2$ with $1$ when finding the derivative with respect to each of $c_1$ and $c_2$, because each will vary independently of the other in the computation of the separate derivatives. I get $S^{-1}(u_1-c_1u_1-c_2u_2)$ for $c_1$. Am I missing something?2011-01-14

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