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I have a matrix kernel function which I am trying to find the derivative to. Function is K = c * exp[-1/2 * (P(X1 - X2))' * P(X1 -X2)] where uppercase are matrices and lower case are scalars (and ' denotes transpose). I'm trying to find dK/dP. I'm pretty rusty on matrix calculus, can anyone give me a hand here?

Thanks

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    (And the factor $1/2$ in my previous comment is not correct)2012-06-13

1 Answers 1

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Let $Z(P)=(P(X_1-X_2))^TP(X_1-X_2)$ If $K$ is viewed as depending only on $P$ and if you differentiate in direction $V$ you get $ D_V K = K*\frac{-1}{2} \left\{\frac{I-e^{-ad_{Z(P)}}}{ad_{Z(p)}} \left[ (V(X_1-X_2))^TP(X_1-X_2) + P(X_1-X_2)^TV(X_1-X_2) \right] \right\}$

The term on the right hand side in curly parenthesis needs explanation. It is the matrix valued function $\frac{I-e^{-ad_{Z(P)}}}{ad_{Z(P))}}$ applied to the term in square brackets. This in turn means

$\frac{I-e^{-ad_{Z}}}{ad_{Z}}[Y] = Y-\frac{[Z,Y]}{2!} + \frac{[Z,[Z,Y]]}{3!} - \ldots$

See, e.g., Chapter 3.3 in Brian C. Halls Book 'Lie Groups, Lie Algebras and Representations for a derivation of the derivative of the exponential.

(Sorry for posting a too simple and wrong answer first, which is true only if $Z$ and $D_VZ $ commute). (I don't like the $\frac{d}{dP}$ notation).