Where $X$ and $C$ are both symmetric positive-definite matrices, and $X^{1/2}$ is the lower-triangular matrix sqare root of $X$. In other words, if $Y=X^{1/2}C{X^{1/2}}'$, what is the effect of a change in the elements of $X$ on those of $Y$? Thanks!!!
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I suggest to first study the derivative $\partial X^{1/2} / \partial x_{ij}$. Since $X$ must lie in the cone of positive definite matrices for $X^{1/2}$ to exist, what does it mean to vary its entry $x_{ij}$ by an amount $\epsilon$? To stay inside the cone, any change of the $(i,j)$-th entry from $x_{ij}$ to $x_{ij}+\epsilon$ for $i \neq j$ should be compensated by a change of the $(j,i)$-th entry from $x_{ji}=x_{ij}^$ to $x_{ji}+\epsilon^$. Think thoroughly about your definition of differentiation. You need to make it precise. – jens Apr 10 '15 at 11:54
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In your problem, it might be that an operation is applied on $Y$ which allows you to ignore these difficulties. For example, if you have $\mathrm{tr}(Y)$ or $\det(\mathrm{I}+Y)$ or something of that kind, you can cyclically shift the matrix factors so as to reunite $X^{1/2}$ and ${X^{1/2}}'$ into a single matrix $X$. – jens Apr 10 '15 at 11:58