| Schur {Matrix} | R Documentation |
Computes the Schur decomposition and eigenvalues of a square matrix; see the BACKGROUND information below.
Schur(x, vectors, ...)
x |
numeric (or complex, in future) square Matrix (inheriting from class
|
vectors |
logical. When |
... |
further arguments passed to or from other methods. |
Based on the Lapack subroutine dgees.
If vectors are TRUE, as per default,
an object of class Schur.
If vectors are FALSE, a list with components
T |
the upper quasi-triangular (square) matrix of the Schur decomposition. |
EValues |
If A is a square matrix, then A = Q T t(Q), where
Q is orthogonal, and T is upper block-triangular
(nearly triangular with either 1 by 1 or 2 by 2 blocks on the
diagonal) where the 2 by 2 blocks correspond to (non-real) complex
eigenvalues.
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.
Anderson, E., et al. (1994). LAPACK User's Guide, 2nd edition, SIAM, Philadelphia.
Schur(Hilbert(9)) # Schur factorization (real eigenvalues)
(A <- Matrix(round(rnorm(5*5, sd = 100)), nrow = 5))
(Sch.A <- Schur(A))
eTA <- eigen(Sch.A@T)
str(SchA <- Schur(A, vectors=FALSE))# no 'T' ==> simple list
stopifnot(all.equal(eTA$values, eigen(A)$values, tol = 1e-13),
all.equal(eTA$values,
local({z <- Sch.A@EValues
z[order(Mod(z), decreasing=TRUE)]}), tol = 1e-13),
identical(SchA$T, Sch.A@T),
identical(SchA$EValues, Sch.A@EValues))