qz.zgeev {QZ}

Generalized Eigenvalues Decomposition for a Complex Matrix

Description

This function call 'zgeev' in Fortran to decompose a 'complex' matrix A.

Usage

  qz.zgeev(A, vl = TRUE, vr = TRUE, LWORK = NULL)

Arguments

A

a 'complex' matrix, dim = c(N, N).

vl

if compute left 'complex' eigen vectors. (U)

vr

if compute right 'complex' eigen vectors. (V)

LWORK

optional, dimension of array WORK for workspace. (>= 2N)

Details

See 'zgeev.f' for all details.

ZGEEV computes for an N-by-N complex non-symmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors.

The right eigenvector v(j) of A satisfies

A * v(j) = lambda(j) * v(j)

where lambda(j) is its eigenvalue. The left eigenvector u(j) of A satisfies

u(j)**H * A = lambda(j) * u(j)**H

where u(j)**H denotes the conjugate transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real.

Value

Return a list contains next:

'W'

original returns from 'zgeev.f'.

'VL'

original returns from 'zgeev.f'.

'VR'

original returns from 'zgeev.f'.

'WORK'

optimal LWORK (for zgeev.f only)

'INFO'

= 0: successful. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: QZ iteration failed.

Extra returns in the list:

'U'

the left eigen vectors.

'V'

the right eigen vectors.

Author(s)

Wei-Chen Chen [e-mail]

References

Anderson, E., et al. (1999) LAPACK User's Guide, 3rd edition, SIAM, Philadelphia.

https://www.netlib.org/lapack/complex16/zgeev.f

https://en.wikipedia.org/wiki/Schur_decomposition

See Also

qz.zgees

Examples

library(QZ, quiet = TRUE)

### https://www.nag.com/lapack-ex/node92.html
A <- exA1$A
ret <- qz.zgeev(A)

# Verify 1
diff.R <- A %*% ret$V - matrix(ret$W, 4, 4, byrow = TRUE) * ret$V
diff.L <- H(ret$U) %*% A - matrix(ret$W, 4, 4) * H(ret$U)
round(diff.R)
round(diff.L)

# Verify 2
round(ret$U %*% H(ret$U))
round(ret$V %*% H(ret$V))