Is it faster to calculate inverses of symmetric matrices as opposed to asymmetric matrices? How? I know there are several methods to inverse or decompose matrices.
I am looking for a comparison of the computational cost of inverting an arbitrary real, symmetric matrix vs a real, asymmetric one.
I am familiar with the choices for optimizing the decomposition of asymmetric matrices. Is there a faster one when you are certain you are handling symmetric ones?
 A: It depends on the structure of the symmetric matrix.  If the matrix $A$ is symmetric positive definite, the answer is generally yes since the cost of a Choleski factorization is $1/3n^3 + O(n^2)$.  For a general square matrix, the cost of an LU factorization is $2/3n^3 + O(n^2)$, so a Choleski factorization is cheaper.  For a general symmetric matrix, we can use a symmetric indefinite factorization, which is more expensive than a Choleski factorization, but some variants have the $1/3n^3$ cost.  What's really different and what can drive up the cost is the pivoting scheme required to keep the scheme stable.  Some pivoting schemes are $O(n^3)$, so then it's not necessarily cheaper than LU, but others are not.  A quick search found a paper called "Performance Optimization of Symmetric Factorization Algorithms" by Silvio Tarca that has some of these numbers.
Of course, in practice, it's sort of hard to tell as there's lots of variations of these algorithms and their ability to play clever tricks to keep the computation in cache varies depending on the variant.  Also, there's a ton of games to be played with sparsity, which also affects these results.
However, in short, I'm pretty sure the raw complexity of the symmetric indefinite factorization, with the right pivoting technique, is more efficient than an LU factorization.
