Q:

How to sort eigenvalues from lowest to highest ? And how to extract the corresponding eigenvector

K={{2.99889*10^6,2.57571*10^6,54.9546,285048.,1.37129*10^6,-4.3079*10^6},{2.57571*10^6,2.99893*10^6,131.169,-285052.,1.3713*10^6,-4.30792*10^6},{54.9546,131.169,3.42441*10^7,-4.76309,-113.584,-24.2506},{285048.,-285052.,-4.76309,429098.,4.45647,-2.74554*10^-10},{1.37129*10^6,1.3713*10^6,-113.584,4.45647,4.25528*10^7,-4.2688*10^6},{-4.3079*10^6,-4.30792*10^6,-24.2506,-2.74554*10^-10,-4.2688*10^6,6.77063*10^6}};
M={{44.649,-2.19291,0.000269222,35.2629,20.3676,-33.3301},{-2.19291,44.6498,-0.000556479,-35.2635,20.3682,-33.3301},{0.000269222,-0.000556479,52.3332,0.0001103,-0.000168249,0.000148546},{35.2629,-35.2635,0.0001103,53.1009,-0.000193321,0},{20.3676,20.3682,-0.000168249,-0.000193321,39.598,-32.9862},{-33.3301,-33.3301,0.000148546,0,-32.9862,52.3816}}
{val,vec}=Eigensystem[{K,M}]

Always I have to sort the results manually it is very tedious . Is there any other way to carry out this with out using revers function. 

A:

The answer, so far as I can tell, is the Reverse function:
Reverse[vec]
Reverse[val]

Alternately, if you want to extract the $n$th smallest eigenvalue and the $n$th smallest eigenvector, you can use
First[Eigenvectors[{K, M}, -n]]
First[Eigenvalues[{K, M}, -n]]

All of this assumes that by "largest" and "smallest", you mean largest & smallest by absolute value.  If all of the eigenvalues are known to be positive, then these are the same thing.  (I assume from your notation that you're doing a normal mode problem, in which case all the eigenvalues should be positive if the system is stable.)

A:

Eigensystem orders eigenvalues and eigenvectors in decending order with respect to absolute value. So Reverse is not always the right answer. But you may employ Ordering:
SeedRandom[1];
A = # + #\[Transpose] &@RandomReal[{-1, 1}, {6, 6}];
B = #.#\[Transpose] &@RandomReal[{-1, 1}, {6, 6}];
{val, vec} = Eigensystem[{A,B}];
i = Ordering[val]
{val[[i]], vec[[i]]}

{2, 3, 5, 6, 4, 1}

