the puzzle of eigenvector and eigenvalue

Dear all
I am so glad the R can provide the efficient calculate about
eigenvector and eigenvalue.
However, i have some puzzle about the procedure of eigen.
Fristly, what kind of procedue does the R utilize such that the eigen
are obtained?
For example, A=matrix(c(1,2,4,3),2,2)
we can define the eigenvalue lamda, such as

   det | 1-lamda     4        |  =0
         |    2        3-lamda  |

then we can obtain the lamda=5 and -1
however, i am interesting in that how does the R to obatin lamda?
By the way, how does the R also obtain the eigenvector?

Basically, i must to know those definition in the R because i have to
utilize eigenvalue and eigenvector to develope SVD (single value
decomposition) by myself. Furthermore, i want to obtain the bi-plot
about my data.
Although someone suggest that i can obtain those result by R (base),
there are exist some tough problems which is difficult to solve it. My
data is too huge to do it. for example, the size of matrix is
400000*800.
So, i think that i should follow whole steps by myself.

Finally, it is also my serious problem in the R, I have three
estimators which work for the same data.
Moreover, i utilize the biplot to realize those tendency. But i find
that the the tendency of third estimator mirror the other two, the
figures are linked below

http://hopjimmy123.pixnet.net/album/photo/108526615#pictop

i believe that the tendency should have the same way with the other
estimator and the major problem is about the eigenvalue, i hope that
someone can give me some idea or provide me some solution about this
problem.
Sincerely