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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.

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    $\begingroup$ Have you looked at this post? scicomp.stackexchange.com/questions/81/… $\endgroup$ – Paul Apr 27 '12 at 19:09
  • $\begingroup$ especially the answer scicomp.stackexchange.com/a/378/1117 of the post Paul linked to contains a kind of answer to your question. Apat from that, it is far too vaguely formulated, as a sensible answer would require writing a book of numerical linear algebra. Read instead classics such as Golub & van Loan, and then ask a specific question! $\endgroup$ – Arnold Neumaier Apr 29 '12 at 16:09
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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?

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To add a bit on the direct solver side, matrix symmetry can be exploited to save you a factor of 2 in memory (regardless of definiteness). Symmetric positive definite is even better, because you don't even need to pivot and can use a BLAS3/GEMM based cholesky decomposition. Symmetric indefinite matrices still save the memory, but should be factored using LDL, which requires 2x2 pivoting to be stable. That yields just BLAS2 speed (I think you can block these algorithms for BLAS3, but you end up with weaker stability guarantees).

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