[R] Schur decomposition? (was: matrix of eigenvalues)

[Spencer Graves]

     Does R have a function for the Schur decomposition?  The 
documentation for library(Matrix) describes a function "Schur", but it 
seems to be missing from the Windows version 0.8-14 (2004-09-14) and 
0.8-15 (2004-10-02). 

     The R 2.0.0 pat documentation for "eigen" refers to 
"http://www.netlib.org/lapack/lug/lapack_lug.html";, and the description 
there for eigen analysis of a non-symmetric matrix says, "This problem 
can be solved via the Schur factorization of A, defined in the real case as
A = ZTZT,

where Z is an orthogonal matrix and T is an upper quasi-triangular 
matrix with 1-by-1 and 2-by-2 diagonal blocks, the 2-by-2 blocks 
corresponding to complex conjugate pairs of eigenvalues of A." 

     Thanks,
     Spencer Graves
Douglas Bates wrote:
Christian Jost wrote:
I thought that the function
eigen(A)
will return a matrix with eigenvectors that are independent of each 
other (thus forming a base and the matrix being invertible). This 
seems not to be the case in the following example
A=matrix(c(1,2,0,1),nrow=2,byrow=T)
eigen(A) ->ev
solve(ev$vectors)

note that I try to get the upper triangular form with eigenvalues on 
the diagonal and (possibly) 1 just atop the eigenvalues to be used to 
solve a linear differential equation
x' = Ax, x(0)=x0
x(t) = P exp(D t) P^-1 x0
where D is this upper triangular form and P is the "passage matrix" 
(not sure about the correct english name) given by a base of 
eigenvectors. So the test would be
solve(ev$vectors) %*% A %*% ev$vectors - D
should be 0

Thanks for any help, Christian.
ps: please copy reply also to my address, my subscription to the 
R-help list seems to have delays

That particular matrix has repeated eigenvalues and a degenerate 
eigenspace.

> A <- matrix(c(1,0,2,1),nc=2)
> A
     [,1] [,2]
[1,]    1    2
[2,]    0    1
> eigen(A)
$values
[1] 1 1
$vectors
     [,1]          [,2]
[1,]    1 -1.000000e+00
[2,]    0  1.110223e-16
______________________________________________
[EMAIL PROTECTED] mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide! 
http://www.R-project.org/posting-guide.html

--
Spencer Graves, PhD, Senior Development Engineer
O:  (408)938-4420;  mobile:  (408)655-4567
______________________________________________
[EMAIL PROTECTED] mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html