For a general matrix A, there are many properties to describe it: symmetric positive definite or indefinite, condition number, spectrum and so on. I am curious about how these properties affect the solving of matrix A for both direct method and iterative method like multigrid, krylov subspace methods. Thanks. 
How do the properties of a matrix affect the linear system solving
linear algebra;linear solver
That's a rather big question :-)I would probably advise you to look up a book on numerical linear algebra - for example Iterative Methods for Sparse Linear Systems by Y. Saad. This is a big and complex subject, with a lot of different quirks to learn.In general, however, low condition numbers is good, because it means that the solution is less sensitive to changes in the right hand side of the system. Positive definiteness is important for a multitude of reasons (and some methods, for example Conjugate Gradient won't work without it). The more you know about your matrix in terms of symmetry, structure and properties, the easier it is to find a linear solver, and more importantly, know how good your results will be.I am sorry if this answer isn't what you hoped for, but whole books could be written on the subject. Perhaps you could narrow the question down a bit - are you solving a particular problem, or are you just out to learn numerical linear algebra?