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How to calculate the gradient with respect to $X$ of: $ \log \mathrm{det}\, X^{-1} $ here $X$ is a positive definite matrix, and det is the determinant of a matrix.

How to calculate this? Or what's the result? Thanks!

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    @Fabian why $\log \det X=tr \log X$? $\log \det X=\log \lambda_1+\cdots+\log \lambda_n$, while $tr \log X=\log X_{11}+\cdots+\log X_{nn}$, where $\lambda$ is an eigenvalue of $X$.2018-12-18

3 Answers 3

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I assume that you are asking for the derivative with respect to the elements of the matrix. In this cases first notice that

$\log \det X^{-1} = \log (\det X)^{-1} = -\log \det X$

and thus

$\frac{\partial}{\partial X_{ij}} \log \det X^{-1} = -\frac{\partial}{\partial X_{ij}} \log \det X = - \frac{1}{\det X} \frac{\partial \det X}{\partial X_{ij}} = - \frac{1}{\det X} \mathrm{adj}(X)_{ji} = - (X^{-1})_{ji}$

since $\mathrm{adj}(X) = \det(X) X^{-1}$ for invertible matrices (where $\mathrm{adj}(X)$ is the adjugate of $X$, see http://en.wikipedia.org/wiki/Adjugate).

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    The $\frac{\partial \det X}{\partial X_{ij}} = \mathrm{adj}(X)_{ji}$ was very non-obvious to me, but can be worked out using the [Jacobi formula](https://en.wikipedia.org/wiki/Jacobi's_formula).2017-01-05
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Or you can check section A.4.1 of the book Stephen Boyd, Lieven Vandenberghe, Convex Optimization for an alternative solution, where they compute the gradient without using the adjugate.

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The simplest is probably to observe that $-\log\det (X+tH) = -\log\det X -\log\det(I+tX^{-1}H) \\= -\log\det X - t \textrm{Tr}(X^{-1}H) + o(t),$

where is used the "obvious" fact that $\det(I+A) = 1+\textrm{Tr}(A)+o(|A|)$ (all the other terms are quadratic expressions of the coefficients of $A$).

Notice that $\textrm{Tr}(X^{-1}H)=(X^{-T},H)$ in the Frobenius scalar product, hence $\nabla [-\log\det(X)] = -X^{-T}$ in this scalar product. (This gives another proof that $\nabla\det (X) = cof(X)$.)

Of course if $X$ is symmetric positive definite then $-X^{-1}$ is also a valid expression. Moreover, one has in this case, for $X,Y$ positive definite, $(-X^{-1}+Y^{-1},X-Y)\ge 0$.