Last comment: Currently, it seems that we don't know more efficiently way to compute the positive or negative eigenfactor for the sum of matrices even though we assume that we know all the eigenfactors for each matrix before the sum. The finite number of matrices under the sum are assumed to be real, symmetric matrices which are diagonalizable.
${A = A^T=VDV^T \; \; \; \; \;...(1)\\ A=A^{+}+A^{-} \; \; \; \; \; \; \; \; \; \; \; \; \; ...(2)\\ A^{+} = \sum_{\lambda_{i}>0}{\lambda_{i}{p}_{i}p_{i}^T}, \; \; \; \; \; \; ...(3)\\ A^{-} = \sum_{\lambda_{i}<0}{\lambda_{i}p_{i}p_{i}^T} \; \; \; \; \; \; \;...(4)\\ \text{(1) is the diagonalization of real, symmetric matrices}\\ \; \; \; \; \; D\text{ is the diagonal matrix containing eigenvalues,}\\ \; \; \; \; \; V\text{ contains column vectors as associated eigenvectors to each eigenvalue } \lambda_{i}\\ \text{(2) is the factorization into positive and negative components }\\ \; \; \; \; \; \text{following the definition in (3) and (4)}\\ \text{(3),(4) define the sum of positive eigenfactors and negative counterpart, }\\ \; \; \; \; \; p_{i} \; \text{is the associated eigenvector for an eigenvalue } \lambda_{i}\\ }$
${ \text{Now, we're interested in taking the positive component as defined in (2) and (3) above }\\ \text{but we do so on a sum of diagonalizable matrices.}\\ }$ ${u \in C, u\text{ is nothing but natural-numbered index to tell }\\ \text{finite number of real, symmetric matrices.}}$
${ \bigg(\sum_{u \in C}{A_{u}}\bigg)^{+}= \bigg(\sum_{u \in C}{\Big(A_{u}^{+}+A_{u}^{-}\Big)}\bigg)^{+} \; \; \; \; \;...(5)\\ =\bigg(\sum_{u \in C}{\Big(V_{u}^{+}D_{u}^{+}V_{u}^{T} +V_{u}^{-}D_{u}^{-}V_{u}^{-T}\Big)}\bigg)^{+} \; \; \; \; \; \; \; \; \;...(6)\\ =\bigg(\sum_{u \in C}{V_{u}^{+}D_{u}^{+}V_{u}^{+T}} +\sum_{u \in C}{V_{u}^{-}D_{u}^{-}V_{u}^{-T}}\bigg)^{+} \; \; \; \; ...(7) }$ ${ \text{(5),(6),(7) are equivalent.}\\ \; \; \; \; \; \text{Is there a way to get the positive eigenfactor after the sum more efficiently}\\ \; \; \; \; \;\text{without taking the full eigendecomposition after the sum to compute}\\ \; \; \; \; \;\text{the positive eigenfactor? We assume we know all the eigendecomposition of each } A_{u}. }$