Special properties in the direct solving of sparse symmetric linear systems In the area of computational solving of large sparse linear systems, some solvers specialize only on symmetric sparse matrices, be it positive definite or indefinite as compared to general (non-symmetric) sparse systems solver. 
What mathematical properties does symmetric sparse matrix possess that makes it computationally more efficient to solve as compared to a nonsymmetric matrix?

A great advantage of the sparse Cholesky factorization for symmetric and positive definite matrices is that you do not need to do pivoting for numerical stability but only focus on the symbolic diagonal pivoting to minimize fill-in. So you can completely separate the symbolic and numeric factorization and reuse the structure of the triangular factor when only the values of the original matrix change but not its structure. This is not the case for symmetric indefinite and nonsymmetric sparse direct solvers where you need to do pivoting not only to reduce fill-in but also for numerical stability reasons. Hence the symbolic and numeric factorization are not completely separable.
