Matrix (version 0.3-3)

schur: Schur Decomposition of a Matrix

Description

Computes the Schur decomposition and eigenvalues of a square matrix.

Usage

schur.Matrix(x, vectors=TRUE, rcond=FALSE, threshold=0)

Arguments

x

numeric or complex square Matrix inheriting from class "Matrix". Missing values (NAs) are not allowed.

vectors

logical. When TRUE (the default), the Schur vectors are computed.

rcond

logical vector of length 2. When rcond[1] is TRUE, the reciprocal condition numbers for the eigenvalues are computed. When rcond[2] is TRUE, the reciprocal condition numbers for the eigenvalues are

threshold

a numeric value. A diagonal block associated with a particular eigenvalue will appear in the top left portion of the Schur decomposition if it is greater than or equal to threshold in magnitude. By default, threshold is zero,

Value

• An object of class c("schur.Matrix", "decomp") whose attributes include the eigenvalues and condition numbers (if requested).

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.

Details

Based on the Lapack functions dgeesx and zgeesx.

References

Anderson, E., et al. (1994). LAPACK User's Guide, 2nd edition, SIAM, Philadelphia.

Examples

A <- Matrix(rnorm( 9*9, sd = 100), nrow = 9)
schur.A <- schur(A)                          # schur factorization
mod.eig <- Mod(attr(schur.A, "eigenvalues")) # eigenvalue modulus
schur.A <- schurmod(schur.A)                 # reordered factorization