Suppose that one has a positive definite hermitian matrix $Q$ and wants to write it as a finite sum of positive definite hermitian matrices, like $Q=\sum_{i=1}^NQ_i$. Is there a "natural" way of (possibly uniquely) decomposing such a matrix in a non-trivial way? By "natural" I mean in the sense that every positive integer can be uniquely decomposed as a product of prime numbers. By trivial I mean decompositions of the form $\sum_{i=1}^Nc_iQ_i$, where $\sum_{i=1}^Nc_i=1$.
There is a similar question answered in this forum, but I am not sure if it answers my question. Could the rank-1 decomposition of $Q$ with the help of its SVD considered such an answer?
Thanks a lot in advance, and I am sorry if the question is too general or vague.
Let $H\in M_n (\mathbb{C})$ be a positive semidefinite Hermitian matrix. Let $r$ be its rank. Then $H$ has $r$ real, positive eigenvalues, while the eigenvalue $\lambda = 0$ has multiplicity $n − r$. There exist $p$ column vectors $X_{\alpha}$ , pairwise orthogonal, such that
$H = X_1 (X_1)^* +\dots+ X_r (X_r)^*$
Finally, $H$ is positive definite if and only if $r = n$ (in which case, $0$ is not an eigenvalue).
Since $H$ is positive Hermitian so $H=UDU^*$ for some unitary $U$, and $D=diag(\lambda_1,\dots,\lambda_r,0,0,\dots,0)$
and define $X_{\alpha}=\sqrt{\lambda_{\alpha}} U_{\alpha}$, $U_{\alpha}$ are collumns of $U$