geigen (version 2.3)

gqz: Generalized Schur decomposition

Description

Computes the generalized eigenvalues and Schur form of a pair of matrices.

Usage

gqz(A, B, sort=c("N","-","+","S","B","R"))

Arguments

A

left hand side matrix.

B

right hand side matrix.

sort

how to sort the generalized eigenvalues. See ‘Details’.

Value

The generalized Schur form for numeric matrices is $$ (A,B) = (Q S Z^T, Q T Z^T)$$ The matrices \(Q\) and \(Z\) are orthogonal. The matrix \(S\) is quasi-upper triangular and the matrix \(T\) is upper triangular. The return value is a list containing the following components

S

generalized Schur form of A.

T

generalized Schur form of B.

sdim

the number of eigenvalues (after sorting) for which the sorting condition is true.

alphar

numerator of the real parts of the eigenvalues (numeric).

alphai

numerator of the imaginary parts of the eigenvalues (numeric).

beta

denominator of the expression for the eigenvalues (numeric).

Q

matrix of left Schur vectors (matrix Q).

Z

matrix of right Schur vectors (matrix Z).

The generalized Schur form for complex matrices is $$(A,B) = (Q S Z^H, Q T Z^H)$$ The matrices \(Q\) and \(Z\) are unitary and the matrices \(S\) and \(T\) are upper triangular. The return value is a list containing the following components

S

generalized Schur form of A.

T

generalized Schur form of B.

sdim

the number of eigenvalues. (after sorting) for which the sorting condition is true.

alpha

numerator of the eigenvalues (complex).

beta

denominator of the eigenvalues (complex).

Q

matrix of left Schur vectors (matrix Q).

Z

matrix of right Schur vectors (matrix Z).

The generalized eigenvalues can be computed by calling function gevalues.

Details

Both matrices must be square. This function provides the solution to the generalized eigenvalue problem defined by $$A x = \lambda Bx$$ If either one of the matrices is complex the other matrix is coerced to be complex.

The sort argument specifies how to order the eigenvalues on the diagonal of the generalized Schur form, where it is noted that non-finite eigenvalues never satisfy any ordering condition (even in the case of a complex infinity). Eigenvalues that are placed in the leading block of the Schur form satisfy

N

unordered.

-

negative real part.

+

positive real part.

S

absolute value < 1.

B

absolute value > 1.

R

imaginary part identical to 0 with a tolerance of 100*machine_precision as determined by Lapack.

References

Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM. Available on-line at http://www.netlib.org/lapack/lug/lapack_lug.html. See the section Eigenvalues, Eigenvectors and Generalized Schur Decomposition (http://www.netlib.org/lapack/lug/node56.html).

See Also

geigen, gevalues

Examples

# NOT RUN {
# Real matrices
# example from NAG: http://www.nag.com/lapack-ex/node116.html
# Find the generalized Schur decomposition with the real eigenvalues ordered to come first

A <- matrix(c(  3.9, 12.5,-34.5,-0.5,
                4.3, 21.5,-47.5, 7.5,
                4.3, 21.5,-43.5, 3.5,
                4.4, 26.0,-46.0, 6.0), nrow=4, byrow=TRUE)

B <- matrix(c( 1.0, 2.0, -3.0, 1.0,
               1.0, 3.0, -5.0, 4.0,
               1.0, 3.0, -4.0, 3.0,
               1.0, 3.0, -4.0, 4.0), nrow=4, byrow=TRUE)

z <- gqz(A, B,"R")
z

# complexify
A <- A+0i
B <- B+0i
z <- gqz(A, B,"R")
z
# }