User:A ringh/sandbox

Geometric mean of complex numbers
...

Geometric mean of positive definite matrices
The geometric mean can be extended to positive-definite matrices, and even to accretive matrices, i.e., matrices with a strictly positive Hermitian part.


 * $$GA^{-1}G = B.$$


 * $$G^{-1} = \frac{2}{\pi} \int_0^\infty \left( tA + t^{-1}B \right)\frac{dt}{t}.$$


 * $$A\#B = A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2} = B^{1/2}(B^{-1/2}AB^{-1/2})^{1/2}B^{1/2}.$$

It is the midpoint on the geodesic connecting $$A$$ and $$B$$ on the smooth manifold of positive definite matrices.

A collection of references needed, also exploring different options for how citations work in Wiki. A sentence. Another sentence. A third sentence. A forth sentence. A fifth sentence. A sixth sentence. A seventh sentence.