Orthogonal projection onto cones in inner product spaces

Question

Let $H_n$ denote the space of $n\times n$ Hermitian matrices. For every $A\in H_n$, using the spectral decompostion of $A$, $$A=\sum_i \lambda_i x_ix_i^*,$$ one can define the positive and negative parts of A by simple formulas: $$A^+ =\sum_{i:\,\lambda_i>0} \lambda_i x_ix_i^*, \;A^- =\sum_{i:\,\lambda_i<0} \lambda_i x_ix_i^*.$$

Now if we denote the convex cone of $n\times n$ positive semidefinite matrices by $C$ and also denote the cone $-C$ by $D$, it can be easily shown that $A^+$ and $A^-$ are the unique Hermitian matrices with the following properties:

1. $A^+\in C,A^- \in D$,
2. $A=A^++A^-$,
3. $(A^+,A^-)=0$.

Here $(\cdot,\cdot)$ is the standard Frobenius real inner product on $H_n$ defined by $(A,B)=\mathrm{tr}(AB)$.

Having all of this, we have an interesting kind of orthogonal decomposition or orthogonal projection in the inner product space $H_n$ with respect to the pair of cones $C,D$, which has obviously many applications in matrix analysis and inequalities.

I wonder does there studied a more general notion of orthogonal projection in inner product spaces which embrace also these cases of projection to cones? What is the characterization of these cones (or pairs of cones) which provide this projection? What are the properties (norm, etc) of the components of projection in this version? Any related reference suggestion are welcome!

Answer 1

For a given Hermitian $A$, the matrix $A^+$ is the positive semi-definite matrix $X$ that minimizes the Frobenius norm $\| X-A\|_{\rm F}$, as well as the spectral norm $\| X-A\|_2$, see section 8.1.1, "Euclidean projection on a proper cone" of Boyd & Vandenberghe. There is also a proof at this MSE post.