- Background:

Let $\mathcal{K}$ be set (convex cone, if you like) of symmetric matrices of order $n$. Each matrix $A \in \mathcal{K}$ can be decomposed in a unique way as $A=A_{+}-A_{-}$, where $A_{+}$ and $A_{-}$ are mutually orthogonal positive definite matrices.

 - The question

Let us define $m(A)=\frac{||A_{-}||}{||A||}$, using the Frobenius norm. I am interested in finding a matrix $A \in \mathcal{K}$ that maximizes $A$.

 - Notes:

My original motivation comes from the case when $\mathcal{K}$ is the set of nonnegative matrices. However, the general version at the very least seems to make sense.

A possible generalization could be to decompose $A$ with respect to a more general cone, instead of the cone of positive definite matrices, using [Moreau decomposition][1].


  [1]: http://www.convexoptimization.com/wikimization/index.php/Moreau%2527s_decomposition_theorem