Assume $A(x)=(a\_{ij}(x))\_{k\times k}$ is a Hermitian matrix function on some manifold $M$, is there any inequality relates the integral of its determinant $\int\_M \det(A)$ and the determinant of its integral $\det(\int\_M A)$? Here $\int\_M A$ is the matrix obtained by integrating each entries, i.e. $\int\_M A=(\int\_M a\_{ij}(x)dx)\_{k\times k}$. Thanks.



Exchange determinant and integral of a matrix-valued function
Of course, there are. Suppose $A(x)$ is semi-positive definite. Then, because of the concavity of $A\mapsto(\det A)^{1/n}$, Jenssen's Inequality gives
$$\frac1{{\rm vol}M}\int\_M(\det A)^{1/n}\le\det\left(\frac1{{\rm vol}M}\int A(x)\right)^{1/n}.$$


On the other hand, if $A$ is a constant $B$ plus the Jacobian matrix of a compactly supported field $\phi$ (for instance if $M$ has no boundary), then the integral and the determinant do commutte:
$$\frac1{{\rm vol}M}\int\_M\det(B+ \nabla\phi)=\det B.$$
There are variations on this theme (quasi-convexity in the sense of Morrey, polyconvexity in the sense of Ball).


