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)