schur computes the Schur decomposition of an \(n \times n\) real matrix
\(\bold{A}\).
Usage
schur(x, vectors = TRUE)
Value
The Schur decomposition of the matrix as computed by LAPACK. The components in
the returned value correspond directly to the values returned by DGEES.
m
a matrix with the same dimensions as x. The upper triangle contains
the \(\bold{M}\) matrix of the decomposition.
values
a vector containing the \(n\) eigenvalues of x, these values
are not ordered.
vectors
an \(n \times n\) matrix whose columns contain the eigenvectors
of x, only available if it is requested.
Arguments
x
a square numeric matrix to be decomposed.
vectors
logical, if TRUE (the default), then Schur vectors are returned.
Details
For an \(n \times n\) real matrix \(\bold{A}\), the Schur decomposition
is given by,
$$\bold{A} = \bold{VMV}^T$$
where \(\bold{V}\) is an orthogonal matrix and \(\bold{M}\) is an upper triangular
matrix. The column vectors \(\bold{V}\) (if requested) are the Schur vectors of
\(\bold{A}\), and \(\bold{M}\) is the Schur form of \(\bold{A}\).
Unsuccessful results from the underlying LAPACK code will result in an error giving
a error code: these can only be interpreted by detailed study of the Fortran code.
References
Anderson. E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A. Sorensen, D. (1999).
LAPACK Users' Guide, 3rd Edition. SIAM.
Golub, G.H., Van Loan, C.F. (1996).
Matrix Computations, 3rd Edition.
John Hopkins University Press.
a <- matrix(c(7,12,-2,-3), ncol = 2)
z <- schur(a)
z # information of Schur decompositionx <- matrix(c(0,0,1,2,1,0,2,2,1), ncol = 3)
z <- schur(x)
z # complex eigenvalues