Schur Decomposition of a Matrix --- Generic Function
DESCRIPTION:
Computes the Schur decomposition (including eigenvalues) of a square matrix.
In S-PLUS, only one class has a method for this function -- the
class from the
library. It is defined as a method to
allow users to easily incorporate customized versions into S-PLUS.
USAGE:
schur(x, ...)
REQUIRED ARGUMENTS:
x
a square matrix. No missing values or IEEE special values are allowed.
OPTIONAL ARGUMENTS:
...
most methods will have additional arguments, for example an argument
indicating whether or not to compute the Schur vectors.
VALUE:
a representation of all or part of the Schur decomposition of
x .
BACKGROUND:
If
A is a square matrix, then
A = Q T t(Q) , where
Q is orthogonal, and
T
is upper quasi-triangular (nearly triangular with either 1 by 1 or 2 by 2
blocks on the diagonal). The eigenvalues of
A are the same as those of
T ,
which are easy to compute. The Schur form is used most often for computing
non-symmetric eigenvalue decompositions, and for computing functions of
matrices such as matrix exponentials.
Golub, G., and Van Loan, C. F. (1989).
Matrix Computations,
2nd edition, Johns Hopkins, Baltimore.
SEE ALSO:
EXAMPLES:
library(Matrix)
x <- Matrix( rnorm(9), 3, 3)
schur(x)